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## Multivariable contour integrals
To solve multivariable contour integrals (i.e. surface integrals, complex volume integrals, and higher order integrals), we must use the divergence theorem. For now, let
$$
\nabla \cdot
$$
be interchangeable with
$$
\operatorname{div}
$$
. These will both serve as the divergence of the vector field denoted as
$$
\mathbf{F}
$$
. This theorem states:
$$
\underbrace{\int \cdots \int_U}_n \operatorname{div}(\mathbf{F}) \, dV = \underbrace{ \oint \cdots \oint_{\partial U} }_{n-1} \mathbf{F} \cdot \mathbf{n} \, dS
$$
In addition, we also need to evaluate
$$
\nabla\cdot \mathbf{F}
$$
where
$$
\nabla \cdot \mathbf{F}
$$
is an alternate notation of
$$
\operatorname{div} (\mathbf{F})
$$
.
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The divergence of any dimension can be described as
$$
\begin{align}
\operatorname{div}(\mathbf{F}) &=\nabla\cdot\mathbf{F}\\
&= \left(\frac{\partial}{\partial u}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}, \dots \right) \cdot (F_u,F_x,F_y,F_z,\dots)\\
&=\left(\frac{\partial F_u}{\partial u} + \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} + \cdots \right)
\end{align}
$$
Example 1
Let the vector field
$$
\mathbf{F}=\sin(2x)\mathbf{e}_x+\sin(2y)\mathbf{e}_y+\sin(2z)\mathbf{e}_z
$$
and be bounded by the following
$$
{0\leq x\leq 1} \quad {0\leq y\leq 3} \quad {-1\leq z\leq 4}
$$
The corresponding double contour integral would be set up as such:
We now evaluate
$$
\nabla\cdot\mathbf{F}
$$
.
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Meanwhile, set up the corresponding triple integral:
$$
\begin{align}
&=\iiint_V \left(\frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}\right) dV\\[6pt]
&=\iiint_V \left(\frac{\partial \sin(2x)}{\partial x} + \frac{\partial \sin(2y)}{\partial y} + \frac{\partial \sin(2z)}{\partial z}\right) dV\\[6pt]
&=\iiint_V 2 \left(\cos(2x) + \cos(2y) + \cos(2z)\right) dV \\[6pt]
&=\int_{0}^{1}\int_{0}^{3}\int_{-1}^{4} 2(\cos(2x)+\cos(2y)+\cos(2z))\,dx\,dy\,dz \\[6pt]
&=\int_{0}^{1}\int_{0}^{3}(10\cos(2y)+\sin(8)+\sin(2)+10\cos(z))\,dy\,dz\\[6pt]
&=\int_{0}^{1}(30\cos(2z)+3\sin(2)+3\sin(8)+5\sin(6))\,dz\\[6pt]
&=18\sin(2)+3\sin(8)+5\sin(6)
\end{align}
$$
Example 2
Let the vector field
$$
\mathbf{F}=u^4\mathbf{e}_u+x^5\mathbf{e}_x+y^6\mathbf{e}_y+z^{-3}\mathbf{e}_z
$$
, and remark that there are 4 parameters in this case.
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Let this vector field be bounded by the following:
$$
{0\leq x\leq 1} \quad {-10\leq y\leq 2\pi} \quad {4\leq z\leq 5} \quad {-1\leq u\leq 3}
$$
To evaluate this, we must utilize the divergence theorem as stated before, and we must evaluate
$$
\nabla\cdot\mathbf{F}
$$
.
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Let
$$
dV = dx \, dy \, dz \, du
$$
$$
\begin{align}
&=\iiiint_V \left(\frac{\partial F_u}{\partial u} + \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}\right)\,dV\\[6pt]
&=\iiiint_V \left(\frac{\partial u^4}{\partial u} + \frac{\partial x^5}{\partial x} + \frac{\partial y^6}{\partial y} + \frac{\partial z^{-3}}{\partial z}\right)\,dV\\[6pt]
&=\iiiint_V {{\frac{4 u^3 z^4 + 5 x^4 z^4 + 5 y^4 z^4 - 3}{z^4}}}\,dV \\[6pt]
&= \iiiint_V {{\frac{4 u^3 z^4 + 5 x^4 z^4 + 5 y^4 z^4 - 3}{z^4}}}\,dV \\[6pt]
&=\int_{0}^{1}\int_{-10}^{2\pi}\int_{4}^{5}\int_{-1}^{3} \frac{4 u^3 z^4 + 5 x^4 z^4 + 5 y^4 z^4 - 3}{z^4}\,dV\\[6pt]
&=\int_{0}^{1}\int_{-10}^{2\pi}\int_{4}^{5}\left(\frac{4(3u^4z^3+3y^6+91z^3+3)}{3z^3}\right)\,dy\,dz\,du\\[6pt]
&=\int_{0}^{1}\int_{-10}^{2\pi}\left(4u^4+\frac{743440}{21}+\frac{4}{z^3}\right)\,dz\,du\\[6pt]
&=\int_{0}^{1} \left(-\frac{1}{2\pi^2}+\frac{1486880\pi}{21}+8\pi u^4+40 u^4+\frac{371720021}{1050}\right)\,du\\[6pt]
&=\frac{371728421}{1050}+\frac{14869136\pi^3-105}{210\pi^2}\\[6pt]
&\approx{576468.77}
\end{align}
$$
Thus, we can evaluate a contour integral with
$$
n=4
$$
.
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|
We can use the same method to evaluate contour integrals for any vector field with
$$
n>4
$$
as well.
## Integral representation
An integral representation of a function is an expression of the function involving a contour integral. Various integral representations are known for many special functions. Integral representations can be important for theoretical reasons, e.g. giving analytic continuation or functional equations, or sometimes for numerical evaluations.
For example, the original definition of the Riemann zeta function via a Dirichlet series,
$$
\zeta(s) = \sum_{n=1}^\infty\frac{1}{n^s},
$$
is valid only for . But
$$
\zeta(s) = - \frac{\Gamma(1 - s)}{2 \pi i} \int_H\frac{(-t)^{s-1}}{e^t - 1} dt ,
$$
where the integration is done over the Hankel contour , is valid for all complex s not equal to 1.
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In nuclear physics and nuclear chemistry, a nuclear reaction is a process in which two nuclei, or a nucleus and an external subatomic particle, collide to produce one or more new nuclides. Thus, a nuclear reaction must cause a transformation of at least one nuclide to another. If a nucleus interacts with another nucleus or particle, they then separate without changing the nature of any nuclide, the process is simply referred to as a type of nuclear scattering, rather than a nuclear reaction.
In principle, a reaction can involve more than two particles colliding, but because the probability of three or more nuclei to meet at the same time at the same place is much less than for two nuclei, such an event is exceptionally rare (see triple alpha process for an example very close to a three-body nuclear reaction). The term "nuclear reaction" may refer either to a change in a nuclide induced by collision with another particle or to a spontaneous change of a nuclide without collision.
Natural nuclear reactions occur in the interaction between cosmic rays and matter, and nuclear reactions can be employed artificially to obtain nuclear energy, at an adjustable rate, on-demand. Nuclear chain reactions in fissionable materials produce induced nuclear fission. Various nuclear fusion reactions of light elements power the energy production of the Sun and stars.
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Nuclear chain reactions in fissionable materials produce induced nuclear fission. Various nuclear fusion reactions of light elements power the energy production of the Sun and stars. Most nuclear reactions (fusion and fission) results in transmutation of nuclei (called also nuclear transmutation).
## History
In 1919, Ernest Rutherford was able to accomplish transmutation of nitrogen into oxygen at the University of Manchester, using alpha particles directed at nitrogen 14N + α → 17O + p. This was the first observation of an induced nuclear reaction, that is, a reaction in which particles from one decay are used to transform another atomic nucleus. Eventually, in 1932 at Cambridge University, a fully artificial nuclear reaction and nuclear transmutation was achieved by Rutherford's colleagues John Cockcroft and Ernest Walton, who used artificially accelerated protons against lithium-7, to split the nucleus into two alpha particles. The feat was popularly known as "splitting the atom", although it was not the modern nuclear fission reaction later (in 1938) discovered in heavy elements by the German scientists Otto Hahn, Lise Meitner, and Fritz Strassmann.
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Eventually, in 1932 at Cambridge University, a fully artificial nuclear reaction and nuclear transmutation was achieved by Rutherford's colleagues John Cockcroft and Ernest Walton, who used artificially accelerated protons against lithium-7, to split the nucleus into two alpha particles. The feat was popularly known as "splitting the atom", although it was not the modern nuclear fission reaction later (in 1938) discovered in heavy elements by the German scientists Otto Hahn, Lise Meitner, and Fritz Strassmann.
## Nuclear reaction equations
Nuclear reactions may be shown in a form similar to chemical equations, for which invariant mass must balance for each side of the equation, and in which transformations of particles must follow certain conservation laws, such as conservation of charge and baryon number (total atomic mass number). An example of this notation follows:
To balance the equation above for mass, charge and mass number, the second nucleus to the right must have atomic number 2 and mass number 4; it is therefore also helium-4. The complete equation therefore reads:
or more simply:
Instead of using the full equations in the style above, in many situations a compact notation is used to describe nuclear reactions.
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An example of this notation follows:
To balance the equation above for mass, charge and mass number, the second nucleus to the right must have atomic number 2 and mass number 4; it is therefore also helium-4. The complete equation therefore reads:
or more simply:
Instead of using the full equations in the style above, in many situations a compact notation is used to describe nuclear reactions. This style of the form A(b,c)D is equivalent to A + b producing c + D. Common light particles are often abbreviated in this shorthand, typically p for proton, n for neutron, d for deuteron, α representing an alpha particle or helium-4, β for beta particle or electron, γ for gamma photon, etc. The reaction above would be written as 6Li(d,α)α.
## Energy conservation
Kinetic energy may be released during the course of a reaction (exothermic reaction) or kinetic energy may have to be supplied for the reaction to take place (endothermic reaction). This can be calculated by reference to a table of very accurate particle rest masses, as follows: according to the reference tables, the nucleus has a standard atomic weight of 6.015 atomic mass units (abbreviated u), the deuterium has 2.014 u, and the helium-4 nucleus has 4.0026 u.
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## Energy conservation
Kinetic energy may be released during the course of a reaction (exothermic reaction) or kinetic energy may have to be supplied for the reaction to take place (endothermic reaction). This can be calculated by reference to a table of very accurate particle rest masses, as follows: according to the reference tables, the nucleus has a standard atomic weight of 6.015 atomic mass units (abbreviated u), the deuterium has 2.014 u, and the helium-4 nucleus has 4.0026 u. Thus:
- the sum of the rest mass of the individual nuclei = 6.015 + 2.014 = 8.029 u;
- the total rest mass on the two helium-nuclei = 2 × 4.0026 = 8.0052 u;
- missing rest mass = 8.029 – 8.0052 = 0.0238 atomic mass units.
In a nuclear reaction, the total (relativistic) energy is conserved. The "missing" rest mass must therefore reappear as kinetic energy released in the reaction; its source is the nuclear binding energy. Using Einstein's mass-energy equivalence formula E = mc2, the amount of energy released can be determined.
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The "missing" rest mass must therefore reappear as kinetic energy released in the reaction; its source is the nuclear binding energy. Using Einstein's mass-energy equivalence formula E = mc2, the amount of energy released can be determined. We first need the energy equivalent of one atomic mass unit:
Hence, the energy released is 0.0238 × 931 MeV = 22.2 MeV.
Expressed differently: the mass is reduced by 0.3%, corresponding to 0.3% of 90 PJ/kg is 270 TJ/kg.
This is a large amount of energy for a nuclear reaction; the amount is so high because the binding energy per nucleon of the helium-4 nucleus is unusually high because the He-4 nucleus is "doubly magic". (The He-4 nucleus is unusually stable and tightly bound for the same reason that the helium atom is inert: each pair of protons and neutrons in He-4 occupies a filled 1s nuclear orbital in the same way that the pair of electrons in the helium atom occupy a filled 1s electron orbital). Consequently, alpha particles appear frequently on the right-hand side of nuclear reactions.
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(The He-4 nucleus is unusually stable and tightly bound for the same reason that the helium atom is inert: each pair of protons and neutrons in He-4 occupies a filled 1s nuclear orbital in the same way that the pair of electrons in the helium atom occupy a filled 1s electron orbital). Consequently, alpha particles appear frequently on the right-hand side of nuclear reactions.
The energy released in a nuclear reaction can appear mainly in one of three ways:
- kinetic energy of the product particles (fraction of the kinetic energy of the charged nuclear reaction products can be directly converted into electrostatic energy);
- emission of very high energy photons, called gamma rays;
- some energy may remain in the nucleus, as a metastable energy level.
When the product nucleus is metastable, this is indicated by placing an asterisk ("*") next to its atomic number. This energy is eventually released through nuclear decay.
A small amount of energy may also emerge in the form of X-rays. Generally, the product nucleus has a different atomic number, and thus the configuration of its electron shells is wrong. As the electrons rearrange themselves and drop to lower energy levels, internal transition X-rays (X-rays with precisely defined emission lines) may be emitted.
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Generally, the product nucleus has a different atomic number, and thus the configuration of its electron shells is wrong. As the electrons rearrange themselves and drop to lower energy levels, internal transition X-rays (X-rays with precisely defined emission lines) may be emitted.
## Q-value and energy balance
In writing down the reaction equation, in a way analogous to a chemical equation, one may, in addition, give the reaction energy on the right side:
For the particular case discussed above, the reaction energy has already been calculated as Q = 22.2 MeV. Hence:
The reaction energy (the "Q-value") is positive for exothermal reactions and negative for endothermal reactions, opposite to the similar expression in chemistry. On the one hand, it is the difference between the sums of kinetic energies on the final side and on the initial side. But on the other hand, it is also the difference between the nuclear rest masses on the initial side and on the final side (in this way, we have calculated the Q-value above).
## Reaction rates
If the reaction equation is balanced, that does not mean that the reaction really occurs. The rate at which reactions occur depends on the energy and the flux of the incident particles, and the reaction cross section.
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## Reaction rates
If the reaction equation is balanced, that does not mean that the reaction really occurs. The rate at which reactions occur depends on the energy and the flux of the incident particles, and the reaction cross section. An example of a large repository of reaction rates is the REACLIB database, as maintained by the Joint Institute for Nuclear Astrophysics.
## Charged vs. uncharged particles
In the initial collision which begins the reaction, the particles must approach closely enough so that the short-range strong force can affect them. As most common nuclear particles are positively charged, this means they must overcome considerable electrostatic repulsion before the reaction can begin. Even if the target nucleus is part of a neutral atom, the other particle must penetrate well beyond the electron cloud and closely approach the nucleus, which is positively charged. Thus, such particles must be first accelerated to high energy, for example by:
- particle accelerators;
- nuclear decay (alpha particles are the main type of interest here since beta and gamma rays are rarely involved in nuclear reactions);
- very high temperatures, on the order of millions of degrees, producing thermonuclear reactions;
- cosmic rays.
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Even if the target nucleus is part of a neutral atom, the other particle must penetrate well beyond the electron cloud and closely approach the nucleus, which is positively charged. Thus, such particles must be first accelerated to high energy, for example by:
- particle accelerators;
- nuclear decay (alpha particles are the main type of interest here since beta and gamma rays are rarely involved in nuclear reactions);
- very high temperatures, on the order of millions of degrees, producing thermonuclear reactions;
- cosmic rays.
Also, since the force of repulsion is proportional to the product of the two charges, reactions between heavy nuclei are rarer, and require higher initiating energy, than those between a heavy and light nucleus; while reactions between two light nuclei are the most common ones.
Neutrons, on the other hand, have no electric charge to cause repulsion, and are able to initiate a nuclear reaction at very low energies. In fact, at extremely low particle energies (corresponding, say, to thermal equilibrium at room temperature), the neutron's de Broglie wavelength is greatly increased, possibly greatly increasing its capture cross-section, at energies close to resonances of the nuclei involved. Thus low-energy neutrons may be even more reactive than high-energy neutrons.
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In fact, at extremely low particle energies (corresponding, say, to thermal equilibrium at room temperature), the neutron's de Broglie wavelength is greatly increased, possibly greatly increasing its capture cross-section, at energies close to resonances of the nuclei involved. Thus low-energy neutrons may be even more reactive than high-energy neutrons.
## Notable types
While the number of possible nuclear reactions is immense, there are several types that are more common, or otherwise notable. Some examples include:
- Fusion reactions – two light nuclei join to form a heavier one, with additional particles (usually protons or neutrons) emitted subsequently.
- Spallation – a nucleus is hit by a particle with sufficient energy and momentum to knock out several small fragments or smash it into many fragments.
- Induced gamma emission belongs to a class in which only photons were involved in creating and destroying states of nuclear excitation.
- Fission reactions – a very heavy nucleus, after absorbing additional light particles (usually neutrons), splits into two or sometimes three pieces. This is an induced nuclear reaction. Spontaneous fission, which occurs without assistance of a neutron, is usually not considered a nuclear reaction. At most, it is not an induced nuclear reaction.
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Spontaneous fission, which occurs without assistance of a neutron, is usually not considered a nuclear reaction. At most, it is not an induced nuclear reaction.
### Direct reactions
An intermediate energy projectile transfers energy or picks up or loses nucleons to the nucleus in a single quick (10−21 second) event. Energy and momentum transfer are relatively small. These are particularly useful in experimental nuclear physics, because the reaction mechanisms are often simple enough to calculate with sufficient accuracy to probe the structure of the target nucleus.
#### Inelastic scattering
Only energy and momentum are transferred.
- (p,p') tests differences between nuclear states.
- (α,α') measures nuclear surface shapes and sizes. Since α particles that hit the nucleus react more violently, elastic and shallow inelastic α scattering are sensitive to the shapes and sizes of the targets, like light scattered from a small black object.
- (e,e') is useful for probing the interior structure. Since electrons interact less strongly than do protons and neutrons, they reach to the centers of the targets and their wave functions are less distorted by passing through the nucleus.
#### Charge-exchange reactions
Energy and charge are transferred between projectile and target.
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Since electrons interact less strongly than do protons and neutrons, they reach to the centers of the targets and their wave functions are less distorted by passing through the nucleus.
#### Charge-exchange reactions
Energy and charge are transferred between projectile and target. Some examples of this kind of reactions are:
- (p,n)
- (3He,t)
#### Nucleon transfer reactions
Usually at moderately low energy, one or more nucleons are transferred between the projectile and target. These are useful in studying outer shell structure of nuclei. Transfer reactions can occur:
- from the projectile to the target - stripping reactions
- from the target to the projectile - pick-up reactions
Examples:
- (α,n) and (α,p) reactions. Some of the earliest nuclear reactions studied involved an alpha particle produced by alpha decay, knocking a nucleon from a target nucleus.
- (d,n) and (d,p) reactions. A deuteron beam impinges on a target; the target nuclei absorb either the neutron or proton from the deuteron. The deuteron is so loosely bound that this is almost the same as proton or neutron capture. A compound nucleus may be formed, leading to additional neutrons being emitted more slowly.
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The deuteron is so loosely bound that this is almost the same as proton or neutron capture. A compound nucleus may be formed, leading to additional neutrons being emitted more slowly. (d,n) reactions are used to generate energetic neutrons.
- The strangeness exchange reaction (K, π) has been used to study hypernuclei.
- The reaction 14N(α,p)17O performed by Rutherford in 1917 (reported 1919), is generally regarded as the first nuclear transmutation experiment.
#### Reactions with neutrons
→ T → 7Li → 14C (n,α)6Li + n → T + α 10B + n → 7Li + α 17O + n → 14C + α 21Ne + n → 18O + α 37Ar + n → 34S + α(n,p)3He + n → T + p 7Be + n → 7Li + p 14N + n → 14C + p 22Na + n → 22Ne + p (n,γ)2H + n → T + γ 13C + n → 14C + γ
Reactions with neutrons are important in nuclear reactors and nuclear weapons.
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The reaction 14N(α,p)17O performed by Rutherford in 1917 (reported 1919), is generally regarded as the first nuclear transmutation experiment.
#### Reactions with neutrons
→ T → 7Li → 14C (n,α)6Li + n → T + α 10B + n → 7Li + α 17O + n → 14C + α 21Ne + n → 18O + α 37Ar + n → 34S + α(n,p)3He + n → T + p 7Be + n → 7Li + p 14N + n → 14C + p 22Na + n → 22Ne + p (n,γ)2H + n → T + γ 13C + n → 14C + γ
Reactions with neutrons are important in nuclear reactors and nuclear weapons. While the best-known neutron reactions are neutron scattering, neutron capture, and nuclear fission, for some light nuclei (especially odd-odd nuclei) the most probable reaction with a thermal neutron is a transfer reaction:
Some reactions are only possible with fast neutrons:
- (n,2n) reactions produce small amounts of protactinium-231 and uranium-232 in the thorium cycle which is otherwise relatively free of highly radioactive actinide products.
- 9Be + n → 2α + 2n can contribute some additional neutrons in the beryllium neutron reflector of a nuclear weapon.
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#### Reactions with neutrons
→ T → 7Li → 14C (n,α)6Li + n → T + α 10B + n → 7Li + α 17O + n → 14C + α 21Ne + n → 18O + α 37Ar + n → 34S + α(n,p)3He + n → T + p 7Be + n → 7Li + p 14N + n → 14C + p 22Na + n → 22Ne + p (n,γ)2H + n → T + γ 13C + n → 14C + γ
Reactions with neutrons are important in nuclear reactors and nuclear weapons. While the best-known neutron reactions are neutron scattering, neutron capture, and nuclear fission, for some light nuclei (especially odd-odd nuclei) the most probable reaction with a thermal neutron is a transfer reaction:
Some reactions are only possible with fast neutrons:
- (n,2n) reactions produce small amounts of protactinium-231 and uranium-232 in the thorium cycle which is otherwise relatively free of highly radioactive actinide products.
- 9Be + n → 2α + 2n can contribute some additional neutrons in the beryllium neutron reflector of a nuclear weapon.
- 7Li + n → T + α + n unexpectedly contributed additional yield in the Bravo, Romeo and Yankee shots of Operation Castle, the three highest-yield nuclear tests conducted by the U.S.
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While the best-known neutron reactions are neutron scattering, neutron capture, and nuclear fission, for some light nuclei (especially odd-odd nuclei) the most probable reaction with a thermal neutron is a transfer reaction:
Some reactions are only possible with fast neutrons:
- (n,2n) reactions produce small amounts of protactinium-231 and uranium-232 in the thorium cycle which is otherwise relatively free of highly radioactive actinide products.
- 9Be + n → 2α + 2n can contribute some additional neutrons in the beryllium neutron reflector of a nuclear weapon.
- 7Li + n → T + α + n unexpectedly contributed additional yield in the Bravo, Romeo and Yankee shots of Operation Castle, the three highest-yield nuclear tests conducted by the U.S.
### Compound nuclear reactions
Either a low-energy projectile is absorbed or a higher energy particle transfers energy to the nucleus, leaving it with too much energy to be fully bound together. On a time scale of about 10−19 seconds, particles, usually neutrons, are "boiled" off. That is, it remains together until enough energy happens to be concentrated in one neutron to escape the mutual attraction. The excited quasi-bound nucleus is called a compound nucleus.
- Low energy (e, e' xn), (γ, xn)
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That is, it remains together until enough energy happens to be concentrated in one neutron to escape the mutual attraction. The excited quasi-bound nucleus is called a compound nucleus.
- Low energy (e, e' xn), (γ, xn) (the xn indicating one or more neutrons), where the gamma or virtual gamma energy is near the giant dipole resonance. These increase the need for radiation shielding around electron accelerators.
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A web shell is a shell-like interface that enables a web server to be remotely accessed, often for the purposes of cyberattacks. A web shell is unique in that a web browser is used to interact with it.
A web shell could be programmed in any programming language that is supported on a server. Web shells are most commonly written in PHP due to the widespread usage of PHP for web applications. Though Active Server Pages, ASP.NET, Python, Perl, Ruby, and Unix shell scripts are also used.
Using network monitoring tools, an attacker can find vulnerabilities that can potentially allow delivery of a web shell. These vulnerabilities are often present in applications that are run on a web server.
An attacker can use a web shell to issue shell commands, perform privilege escalation on the web server, and the ability to upload, delete, download, and execute files to and from the web server.
## General usage
Web shells are used in attacks mostly because they are multi-purpose and difficult to detect.
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An attacker can use a web shell to issue shell commands, perform privilege escalation on the web server, and the ability to upload, delete, download, and execute files to and from the web server.
## General usage
Web shells are used in attacks mostly because they are multi-purpose and difficult to detect. They are commonly used for:
- Data theft
- Infecting website visitors (watering hole attacks)
- Website defacement by modifying files with a malicious intent
- Launch distributed denial-of-service (DDoS) attacks
- To relay commands inside the network which is inaccessible over the Internet
- To use as command and control base, for example as a bot in a botnet system or in way to compromise the security of additional external networks.
Web shells give hackers the ability to steal information, corrupt data, and upload malwares that are more damaging to a system. The issue increasingly escalates when hackers employ compromised servers to infiltrate a system and jeopardize additional machines. Web shells are also a way that malicious individuals target a variety of industries, including government, financial, and defense through cyber espionage. One of the very well known web shells used in this manner is known as “China Chopper.”
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Web shells are also a way that malicious individuals target a variety of industries, including government, financial, and defense through cyber espionage. One of the very well known web shells used in this manner is known as “China Chopper.”
## Delivery of web shells
Web shells are installed through vulnerabilities in web application or weak server security configuration including the following:
- SQL injection;
- Vulnerabilities in applications and services (e.g. web server software such as NGINX or content management system applications such as WordPress);
- File processing and uploading vulnerabilities, which can be mitigated by e.g. limiting the file types that can be uploaded;
- Remote file inclusion (RFI) and local file inclusion (LFI) vulnerabilities;
- Remote code execution;
- Exposed administration interfaces;
An attacker may also modify (spoof) the `Content-Type` header to be sent by the attacker in a file upload to bypass improper file validation (validation using MIME type sent by the client), which will result in a successful upload of the attacker's shell.
## Example
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## Delivery of web shells
Web shells are installed through vulnerabilities in web application or weak server security configuration including the following:
- SQL injection;
- Vulnerabilities in applications and services (e.g. web server software such as NGINX or content management system applications such as WordPress);
- File processing and uploading vulnerabilities, which can be mitigated by e.g. limiting the file types that can be uploaded;
- Remote file inclusion (RFI) and local file inclusion (LFI) vulnerabilities;
- Remote code execution;
- Exposed administration interfaces;
An attacker may also modify (spoof) the `Content-Type` header to be sent by the attacker in a file upload to bypass improper file validation (validation using MIME type sent by the client), which will result in a successful upload of the attacker's shell.
## Example
The following is a simple example of a web shell written in PHP that executes and outputs the result of a shell command:
```php
<?=`$_GET[x]`?>
```
Assuming the filename is `example.php`, an example that would output the contents of the `/etc/passwd` file is shown below:
https://example.com/example.php?x=cat%20%2Fetc%2Fpasswd
The above request will take the value of the `x` parameter of the query string, sending the following shell command:
```shell
cat /etc/passwd
```
This could have been prevented if the shell functions of PHP were disabled so that arbitrary shell commands cannot be executed from PHP.
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## Example
The following is a simple example of a web shell written in PHP that executes and outputs the result of a shell command:
```php
<?=`$_GET[x]`?>
```
Assuming the filename is `example.php`, an example that would output the contents of the `/etc/passwd` file is shown below:
https://example.com/example.php?x=cat%20%2Fetc%2Fpasswd
The above request will take the value of the `x` parameter of the query string, sending the following shell command:
```shell
cat /etc/passwd
```
This could have been prevented if the shell functions of PHP were disabled so that arbitrary shell commands cannot be executed from PHP.
## Prevention and mitigation
A web shell is usually installed by taking advantage of vulnerabilities present in the web server's software. That is why removal of these vulnerabilities is important to avoid the potential risk of a compromised web server.
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## Prevention and mitigation
A web shell is usually installed by taking advantage of vulnerabilities present in the web server's software. That is why removal of these vulnerabilities is important to avoid the potential risk of a compromised web server.
The following are security measures for preventing the installation of a web shell:
- Regularly update the applications and the host server's operating system to ensure immunity from known bugs
- Deploying a demilitarized zone (DMZ) between the web facing servers and the internal networks
- Secure configuration of the web server
- Closing or blocking ports and services which are not used
- Using user input data validation to limit local and remote file inclusion vulnerabilities
- Use a reverse proxy service to restrict the administrative URL's to known legitimate ones
- Frequent vulnerability scan to detect areas of risk and conduct regular scans using web security software (this does not prevent zero day attacks)
- Deploy a firewall
- Disable directory browsing
- Not using default passwords
## Detection
Web shells can be easily modified, so it's not easy to detect web shells and antivirus software are often not able to detect web shells.
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https://en.wikipedia.org/wiki/Web_shell
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The following are security measures for preventing the installation of a web shell:
- Regularly update the applications and the host server's operating system to ensure immunity from known bugs
- Deploying a demilitarized zone (DMZ) between the web facing servers and the internal networks
- Secure configuration of the web server
- Closing or blocking ports and services which are not used
- Using user input data validation to limit local and remote file inclusion vulnerabilities
- Use a reverse proxy service to restrict the administrative URL's to known legitimate ones
- Frequent vulnerability scan to detect areas of risk and conduct regular scans using web security software (this does not prevent zero day attacks)
- Deploy a firewall
- Disable directory browsing
- Not using default passwords
## Detection
Web shells can be easily modified, so it's not easy to detect web shells and antivirus software are often not able to detect web shells.
The following are common indicators that a web shell is present on a web server:
- Abnormal high web server usage (due to heavy downloading and uploading by the attacker);
- Files with an abnormal timestamp (e.g. newer than the last modification date);
- Unknown files in a web server;
- Files having dubious references, for example, `cmd.exe` or `eval`;
- Unknown connections in the logs of web server
For example, a file generating suspicious traffic (e.g. a PNG file requesting with POST parameters).
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## Detection
Web shells can be easily modified, so it's not easy to detect web shells and antivirus software are often not able to detect web shells.
The following are common indicators that a web shell is present on a web server:
- Abnormal high web server usage (due to heavy downloading and uploading by the attacker);
- Files with an abnormal timestamp (e.g. newer than the last modification date);
- Unknown files in a web server;
- Files having dubious references, for example, `cmd.exe` or `eval`;
- Unknown connections in the logs of web server
For example, a file generating suspicious traffic (e.g. a PNG file requesting with POST parameters). Dubious logins from DMZ servers to internal sub-nets and vice versa.
Web shells may also contain a login form, which is often disguised as an error page.
Using web shells, adversaries can modify the .htaccess file (on servers running the Apache HTTP Server software) on web servers to redirect search engine requests to the web page with malware or spam. Often web shells detect the user-agent and the content presented to the search engine spider is different from that presented to the user's browser. To find a web shell a user-agent change of the crawler bot is usually required.
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Often web shells detect the user-agent and the content presented to the search engine spider is different from that presented to the user's browser. To find a web shell a user-agent change of the crawler bot is usually required. Once the web shell is identified, it can be deleted easily.
Analyzing the web server's log could specify the exact location of the web shell. Legitimate users/visitor usually have different user-agents and referers, on the other hand, a web shell is usually only visited by the attacker, therefore have very few variants of user-agent strings.
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In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a continuous function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically as . The process of solving for antiderivatives is called antidifferentiation (or indefinite integration), and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are often denoted by capital Roman letters such as and .
Antiderivatives are related to definite integrals through the second fundamental theorem of calculus: the definite integral of a function over a closed interval where the function is Riemann integrable is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.
In physics, antiderivatives arise in the context of rectilinear motion (e.g., in explaining the relationship between position, velocity and acceleration). The discrete equivalent of the notion of antiderivative is antidifference.
## Examples
The function
$$
F(x) = \tfrac{x^3}{3}
$$
is an antiderivative of
$$
f(x) = x^2
$$
, since the derivative of
$$
\tfrac{x^3}{3}
$$
is
$$
x^2
$$
.
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The discrete equivalent of the notion of antiderivative is antidifference.
## Examples
The function
$$
F(x) = \tfrac{x^3}{3}
$$
is an antiderivative of
$$
f(x) = x^2
$$
, since the derivative of
$$
\tfrac{x^3}{3}
$$
is
$$
x^2
$$
. Since the derivative of a constant is zero,
$$
x^2
$$
will have an infinite number of antiderivatives, such as
$$
\tfrac{x^3}{3}, \tfrac{x^3}{3}+1, \tfrac{x^3}{3}-2
$$
, etc. Thus, all the antiderivatives of
$$
x^2
$$
can be obtained by changing the value of in
$$
F(x) = \tfrac{x^3}{3}+C
$$
, where is an arbitrary constant known as the constant of integration. The graphs of antiderivatives of a given function are vertical translations of each other, with each graph's vertical location depending upon the value .
More generally, the power function
$$
f(x) = x^n
$$
has antiderivative _ BLOCK9_ if , and
$$
F(x) = \ln |x| + C
$$
if .
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BLOCK9_ if , and
$$
F(x) = \ln |x| + C
$$
if .
In physics, the integration of acceleration yields velocity plus a constant. The constant is the initial velocity term that would be lost upon taking the derivative of velocity, because the derivative of a constant term is zero. This same pattern applies to further integrations and derivatives of motion (position, velocity, acceleration, and so on). Thus, integration produces the relations of acceleration, velocity and displacement:
$$
\begin{align}
\int a \, \mathrm{d}t &= v + C \\
\int v \, \mathrm{d}t &= s + C
\end{align}
$$
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This same pattern applies to further integrations and derivatives of motion (position, velocity, acceleration, and so on). Thus, integration produces the relations of acceleration, velocity and displacement:
$$
\begin{align}
\int a \, \mathrm{d}t &= v + C \\
\int v \, \mathrm{d}t &= s + C
\end{align}
$$
## Uses and properties
Antiderivatives can be used to compute definite integrals, using the fundamental theorem of calculus: if is an antiderivative of the continuous function over the interval
$$
[a,b]
$$
, then:
$$
\int_a^b f(x)\,\mathrm{d}x = F(b) - F(a).
$$
Because of this, each of the infinitely many antiderivatives of a given function may be called the "indefinite integral" of f and written using the integral symbol with no bounds:
$$
\int f(x)\,\mathrm{d}x.
$$
If is an antiderivative of , and the function is defined on some interval, then every other antiderivative of differs from by a constant: there exists a number such that
$$
G(x) = F(x)+c
$$
for all . is called the constant of integration.
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Thus, integration produces the relations of acceleration, velocity and displacement:
$$
\begin{align}
\int a \, \mathrm{d}t &= v + C \\
\int v \, \mathrm{d}t &= s + C
\end{align}
$$
## Uses and properties
Antiderivatives can be used to compute definite integrals, using the fundamental theorem of calculus: if is an antiderivative of the continuous function over the interval
$$
[a,b]
$$
, then:
$$
\int_a^b f(x)\,\mathrm{d}x = F(b) - F(a).
$$
Because of this, each of the infinitely many antiderivatives of a given function may be called the "indefinite integral" of f and written using the integral symbol with no bounds:
$$
\int f(x)\,\mathrm{d}x.
$$
If is an antiderivative of , and the function is defined on some interval, then every other antiderivative of differs from by a constant: there exists a number such that
$$
G(x) = F(x)+c
$$
for all . is called the constant of integration. If the domain of is a disjoint union of two or more (open) intervals, then a different constant of integration may be chosen for each of the intervals.
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## Uses and properties
Antiderivatives can be used to compute definite integrals, using the fundamental theorem of calculus: if is an antiderivative of the continuous function over the interval
$$
[a,b]
$$
, then:
$$
\int_a^b f(x)\,\mathrm{d}x = F(b) - F(a).
$$
Because of this, each of the infinitely many antiderivatives of a given function may be called the "indefinite integral" of f and written using the integral symbol with no bounds:
$$
\int f(x)\,\mathrm{d}x.
$$
If is an antiderivative of , and the function is defined on some interval, then every other antiderivative of differs from by a constant: there exists a number such that
$$
G(x) = F(x)+c
$$
for all . is called the constant of integration. If the domain of is a disjoint union of two or more (open) intervals, then a different constant of integration may be chosen for each of the intervals. For instance
$$
F(x) = \begin{cases}
-\dfrac{1}{x} + c_1 & x<0 \\[1ex]
-\dfrac{1}{x} + c_2 & x>0
\end{cases}
$$
is the most general antiderivative of
$$
f(x)=1/x^2
$$
on its natural domain
$$
(-\infty,0) \cup (0,\infty).
$$
Every continuous function has an antiderivative, and one antiderivative is given by the definite integral of with variable upper
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If the domain of is a disjoint union of two or more (open) intervals, then a different constant of integration may be chosen for each of the intervals. For instance
$$
F(x) = \begin{cases}
-\dfrac{1}{x} + c_1 & x<0 \\[1ex]
-\dfrac{1}{x} + c_2 & x>0
\end{cases}
$$
is the most general antiderivative of
$$
f(x)=1/x^2
$$
on its natural domain
$$
(-\infty,0) \cup (0,\infty).
$$
Every continuous function has an antiderivative, and one antiderivative is given by the definite integral of with variable upper boundary:
$$
F(x) = \int_a^x f(t)\,\mathrm{d}t ~,
$$
for any in the domain of . Varying the lower boundary produces other antiderivatives, but not necessarily all possible antiderivatives. This is another formulation of the fundamental theorem of calculus.
There are many elementary functions whose antiderivatives, even though they exist, cannot be expressed in terms of elementary functions. Elementary functions are polynomials, exponential functions, logarithms, trigonometric functions, inverse trigonometric functions and their combinations under composition and linear combination.
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There are many elementary functions whose antiderivatives, even though they exist, cannot be expressed in terms of elementary functions. Elementary functions are polynomials, exponential functions, logarithms, trigonometric functions, inverse trigonometric functions and their combinations under composition and linear combination. Examples of these nonelementary integrals are
- the error function
$$
\int e^{-x^2}\,\mathrm{d}x,
$$
- the Fresnel function
$$
\int \sin x^2\,\mathrm{d}x,
$$
- the sine integral
$$
\int \frac{\sin x}{x}\,\mathrm{d}x,
$$
- the logarithmic integral function
$$
\int\frac{1}{\log x}\,\mathrm{d}x,
$$
and
- sophomore's dream
$$
\int x^{x}\,\mathrm{d}x.
$$
For a more detailed discussion, see also Differential Galois theory.
## Techniques of integration
Finding antiderivatives of elementary functions is often considerably harder than finding their derivatives (indeed, there is no pre-defined method for computing indefinite integrals). For some elementary functions, it is impossible to find an antiderivative in terms of other elementary functions.
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## Techniques of integration
Finding antiderivatives of elementary functions is often considerably harder than finding their derivatives (indeed, there is no pre-defined method for computing indefinite integrals). For some elementary functions, it is impossible to find an antiderivative in terms of other elementary functions. To learn more, see elementary functions and nonelementary integral.
There exist many properties and techniques for finding antiderivatives. These include, among others:
- The linearity of integration (which breaks complicated integrals into simpler ones)
- Integration by substitution, often combined with trigonometric identities or the natural logarithm
- The inverse chain rule method (a special case of integration by substitution)
- Integration by parts (to integrate products of functions)
- Inverse function integration (a formula that expresses the antiderivative of the inverse of an invertible and continuous function , in terms of and the antiderivative of ).
-
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There exist many properties and techniques for finding antiderivatives. These include, among others:
- The linearity of integration (which breaks complicated integrals into simpler ones)
- Integration by substitution, often combined with trigonometric identities or the natural logarithm
- The inverse chain rule method (a special case of integration by substitution)
- Integration by parts (to integrate products of functions)
- Inverse function integration (a formula that expresses the antiderivative of the inverse of an invertible and continuous function , in terms of and the antiderivative of ).
- The method of partial fractions in integration (which allows us to integrate all rational functions—fractions of two polynomials)
- The Risch algorithm
- Additional techniques for multiple integrations (see for instance double integrals, polar coordinates, the Jacobian and the Stokes' theorem)
- Numerical integration (a technique for approximating a definite integral when no elementary antiderivative exists, as in the case of )
- Algebraic manipulation of integrand (so that other integration techniques, such as integration by substitution, may be used)
- Cauchy formula for repeated integration (to calculate the -times antiderivative of a function) _ BLOCK0_Computer algebra systems can be used to automate some or all of the work involved in the symbolic techniques above, which is particularly useful when the algebraic manipulations involved are very complex or lengthy.
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The method of partial fractions in integration (which allows us to integrate all rational functions—fractions of two polynomials)
- The Risch algorithm
- Additional techniques for multiple integrations (see for instance double integrals, polar coordinates, the Jacobian and the Stokes' theorem)
- Numerical integration (a technique for approximating a definite integral when no elementary antiderivative exists, as in the case of )
- Algebraic manipulation of integrand (so that other integration techniques, such as integration by substitution, may be used)
- Cauchy formula for repeated integration (to calculate the -times antiderivative of a function) _ BLOCK0_Computer algebra systems can be used to automate some or all of the work involved in the symbolic techniques above, which is particularly useful when the algebraic manipulations involved are very complex or lengthy. Integrals which have already been derived can be looked up in a table of integrals.
## Of non-continuous functions
Non-continuous functions can have antiderivatives. While there are still open questions in this area, it is known that:
- Some highly pathological functions with large sets of discontinuities may nevertheless have antiderivatives.
- In some cases, the antiderivatives of such pathological functions may be found by Riemann integration, while in other cases these functions are not Riemann integrable.
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While there are still open questions in this area, it is known that:
- Some highly pathological functions with large sets of discontinuities may nevertheless have antiderivatives.
- In some cases, the antiderivatives of such pathological functions may be found by Riemann integration, while in other cases these functions are not Riemann integrable.
Assuming that the domains of the functions are open intervals:
- A necessary, but not sufficient, condition for a function to have an antiderivative is that have the intermediate value property. That is, if is a subinterval of the domain of and is any real number between and , then there exists a between and such that . This is a consequence of Darboux's theorem.
- The set of discontinuities of must be a meagre set. This set must also be an F-sigma set (since the set of discontinuities of any function must be of this type). Moreover, for any meagre F-sigma set, one can construct some function having an antiderivative, which has the given set as its set of discontinuities.
- If has an antiderivative, is bounded on closed finite subintervals of the domain and has a set of discontinuities of Lebesgue measure 0, then an antiderivative may be found by integration in the sense of Lebesgue.
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Moreover, for any meagre F-sigma set, one can construct some function having an antiderivative, which has the given set as its set of discontinuities.
- If has an antiderivative, is bounded on closed finite subintervals of the domain and has a set of discontinuities of Lebesgue measure 0, then an antiderivative may be found by integration in the sense of Lebesgue. In fact, using more powerful integrals like the Henstock–Kurzweil integral, every function for which an antiderivative exists is integrable, and its general integral coincides with its antiderivative.
- If has an antiderivative on a closed interval
$$
[a,b]
$$
, then for any choice of partition
$$
a=x_0 <x_1 <x_2 <\dots <x_n=b,
$$
if one chooses sample points
$$
x_i^*\in[x_{i-1},x_i]
$$
as specified by the mean value theorem, then the corresponding Riemann sum telescopes to the value
$$
F(b)-F(a)
$$
.
$$
\begin{align}
\sum_{i=1}^n f(x_i^*)(x_i-x_{i-1}) & = \sum_{i=1}^n [F(x_i)-F(x_{i-1})] \\
& = F(x_n)-F(x_0) = F(b)-F(a)
\end{align}
$$
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- If has an antiderivative, is bounded on closed finite subintervals of the domain and has a set of discontinuities of Lebesgue measure 0, then an antiderivative may be found by integration in the sense of Lebesgue. In fact, using more powerful integrals like the Henstock–Kurzweil integral, every function for which an antiderivative exists is integrable, and its general integral coincides with its antiderivative.
- If has an antiderivative on a closed interval
$$
[a,b]
$$
, then for any choice of partition
$$
a=x_0 <x_1 <x_2 <\dots <x_n=b,
$$
if one chooses sample points
$$
x_i^*\in[x_{i-1},x_i]
$$
as specified by the mean value theorem, then the corresponding Riemann sum telescopes to the value
$$
F(b)-F(a)
$$
.
$$
\begin{align}
\sum_{i=1}^n f(x_i^*)(x_i-x_{i-1}) & = \sum_{i=1}^n [F(x_i)-F(x_{i-1})] \\
& = F(x_n)-F(x_0) = F(b)-F(a)
\end{align}
$$
However, if is unbounded, or if is bounded but the set of discontinuities of has positive Lebesgue measure, a different choice of sample points
$$
x_i^*
$$
may give a significantly different value for the Riemann sum, no matter how fine the partition.
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In fact, using more powerful integrals like the Henstock–Kurzweil integral, every function for which an antiderivative exists is integrable, and its general integral coincides with its antiderivative.
- If has an antiderivative on a closed interval
$$
[a,b]
$$
, then for any choice of partition
$$
a=x_0 <x_1 <x_2 <\dots <x_n=b,
$$
if one chooses sample points
$$
x_i^*\in[x_{i-1},x_i]
$$
as specified by the mean value theorem, then the corresponding Riemann sum telescopes to the value
$$
F(b)-F(a)
$$
.
$$
\begin{align}
\sum_{i=1}^n f(x_i^*)(x_i-x_{i-1}) & = \sum_{i=1}^n [F(x_i)-F(x_{i-1})] \\
& = F(x_n)-F(x_0) = F(b)-F(a)
\end{align}
$$
However, if is unbounded, or if is bounded but the set of discontinuities of has positive Lebesgue measure, a different choice of sample points
$$
x_i^*
$$
may give a significantly different value for the Riemann sum, no matter how fine the partition. See Example 4 below.
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However, if is unbounded, or if is bounded but the set of discontinuities of has positive Lebesgue measure, a different choice of sample points
$$
x_i^*
$$
may give a significantly different value for the Riemann sum, no matter how fine the partition. See Example 4 below.
### Some examples
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## Basic formulae
- If
$$
{\mathrm{d} \over \mathrm{d}x} f(x) = g(x)
$$
, then
$$
\int g(x) \mathrm{d}x = f(x) + C
$$
.
-
$$
\int 1\ \mathrm{d}x = x + C
$$
-
$$
\int a\ \mathrm{d}x = ax + C
$$
-
$$
\int x^n \mathrm{d}x = \frac{x^{n+1}}{n+1} + C;\ n \neq -1
$$
-
$$
\int \sin{x}\ \mathrm{d}x = -\cos{x} + C
$$
-
$$
\int \cos{x}\ \mathrm{d}x = \sin{x} + C
$$
-
$$
\int \sec^2{x}\ \mathrm{d}x = \tan{x} + C
$$
-
$$
\int \csc^2{x}\ \mathrm{d}x = -\cot{x} + C
$$
-
$$
\int \sec{x}\tan{x}\ \mathrm{d}x = \sec{x} + C
$$
-
$$
\int \csc{x}\cot{x}\ \mathrm{d}x = -\csc{x} + C
$$
-
$$
\int \frac{1}{x}\ \mathrm{d}x = \ln|x| + C
$$
-
$$
\int \mathrm{e}^{x} \mathrm{d}x = \mathrm{e}^{x} + C
$$
-
$$
\int a^{x} \mathrm{d}x = \frac{a^{x}}{\ln a} + C;\ a > 0,\ a \neq 1
$$
-
$$
\int \frac{1}\sqrt{a^2 - x^2}\ \mathrm{d}x = \arcsin\left(\frac{x}{a}\right) + C
$$
-
$$
\int \frac{1}{a^2 + x^2}\ \mathrm{d}x = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C
$$
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In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualizing and understanding relativistic effects, such as how different observers perceive where and when events occur.
Until the turn of the 20th century, the assumption had been that the three-dimensional geometry of the universe (its description in terms of locations, shapes, distances, and directions) was distinct from time (the measurement of when events occur within the universe). However, space and time took on new meanings with the Lorentz transformation and special theory of relativity.
In 1908, Hermann Minkowski presented a geometric interpretation of special relativity that fused time and the three spatial dimensions into a single four-dimensional continuum now known as Minkowski space. This interpretation proved vital to the general theory of relativity, wherein spacetime is curved by mass and energy.
## Fundamentals
### Definitions
Non-relativistic classical mechanics treats time as a universal quantity of measurement that is uniform throughout, is separate from space, and is agreed on by all observers. Classical mechanics assumes that time has a constant rate of passage, independent of the observer's state of motion, or anything external. It assumes that space is Euclidean: it assumes that space follows the geometry of common sense.
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Classical mechanics assumes that time has a constant rate of passage, independent of the observer's state of motion, or anything external. It assumes that space is Euclidean: it assumes that space follows the geometry of common sense.
In the context of special relativity, time cannot be separated from the three dimensions of space, because the observed rate at which time passes for an object depends on the object's velocity relative to the observer. General relativity provides an explanation of how gravitational fields can slow the passage of time for an object as seen by an observer outside the field.
In ordinary space, a position is specified by three numbers, known as dimensions. In the Cartesian coordinate system, these are often called x, y and z. A point in spacetime is called an event, and requires four numbers to be specified: the three-dimensional location in space, plus the position in time (Fig. 1). An event is represented by a set of coordinates x, y, z and t. Spacetime is thus four-dimensional.
Unlike the analogies used in popular writings to explain events, such as firecrackers or sparks, mathematical events have zero duration and represent a single point in spacetime. Although it is possible to be in motion relative to the popping of a firecracker or a spark, it is not possible for an observer to be in motion relative to an event.
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Unlike the analogies used in popular writings to explain events, such as firecrackers or sparks, mathematical events have zero duration and represent a single point in spacetime. Although it is possible to be in motion relative to the popping of a firecracker or a spark, it is not possible for an observer to be in motion relative to an event.
The path of a particle through spacetime can be considered to be a sequence of events. The series of events can be linked together to form a curve that represents the particle's progress through spacetime. That path is called the particle's world line.
Mathematically, spacetime is a manifold, which is to say, it appears locally "flat" near each point in the same way that, at small enough scales, the surface of a globe appears to be flat. A scale factor,
$$
c
$$
(conventionally called the speed-of-light) relates distances measured in space to distances measured in time. The magnitude of this scale factor (nearly in space being equivalent to one second in time), along with the fact that spacetime is a manifold, implies that at ordinary, non-relativistic speeds and at ordinary, human-scale distances, there is little that humans might observe that is noticeably different from what they might observe if the world were Euclidean.
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A scale factor,
$$
c
$$
(conventionally called the speed-of-light) relates distances measured in space to distances measured in time. The magnitude of this scale factor (nearly in space being equivalent to one second in time), along with the fact that spacetime is a manifold, implies that at ordinary, non-relativistic speeds and at ordinary, human-scale distances, there is little that humans might observe that is noticeably different from what they might observe if the world were Euclidean. It was only with the advent of sensitive scientific measurements in the mid-1800s, such as the Fizeau experiment and the Michelson–Morley experiment, that puzzling discrepancies began to be noted between observation versus predictions based on the implicit assumption of Euclidean space.
In special relativity, an observer will, in most cases, mean a frame of reference from which a set of objects or events is being measured. This usage differs significantly from the ordinary English meaning of the term.
### Reference frames
are inherently nonlocal constructs, and according to this usage of the term, it does not make sense to speak of an observer as having a location.
In Fig. 1-1, imagine that the frame under consideration is equipped with a dense lattice of clocks, synchronized within this reference frame, that extends indefinitely throughout the three dimensions of space. Any specific location within the lattice is not important.
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In Fig. 1-1, imagine that the frame under consideration is equipped with a dense lattice of clocks, synchronized within this reference frame, that extends indefinitely throughout the three dimensions of space. Any specific location within the lattice is not important. The latticework of clocks is used to determine the time and position of events taking place within the whole frame. The term observer refers to the whole ensemble of clocks associated with one inertial frame of reference.
In this idealized case, every point in space has a clock associated with it, and thus the clocks register each event instantly, with no time delay between an event and its recording. A real observer will see a delay between the emission of a signal and its detection due to the speed of light. To synchronize the clocks, in the data reduction following an experiment, the time when a signal is received will be corrected to reflect its actual time were it to have been recorded by an idealized lattice of clocks.
In many books on special relativity, especially older ones, the word "observer" is used in the more ordinary sense of the word. It is usually clear from context which meaning has been adopted.
Physicists distinguish between what one measures or observes, after one has factored out signal propagation delays, versus what one visually sees without such corrections. Failing to understand the difference between what one measures and what one sees is the source of much confusion among students of relativity.
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Physicists distinguish between what one measures or observes, after one has factored out signal propagation delays, versus what one visually sees without such corrections. Failing to understand the difference between what one measures and what one sees is the source of much confusion among students of relativity.
## History
By the mid-1800s, various experiments such as the observation of the Arago spot and differential measurements of the speed of light in air versus water were considered to have proven the wave nature of light as opposed to a corpuscular theory. Propagation of waves was then assumed to require the existence of a waving medium; in the case of light waves, this was considered to be a hypothetical luminiferous aether. The various attempts to establish the properties of this hypothetical medium yielded contradictory results. For example, the Fizeau experiment of 1851, conducted by French physicist Hippolyte Fizeau, demonstrated that the speed of light in flowing water was less than the sum of the speed of light in air plus the speed of the water by an amount dependent on the water's index of refraction.
Among other issues, the dependence of the partial aether-dragging implied by this experiment on the index of refraction (which is dependent on wavelength) led to the unpalatable conclusion that aether simultaneously flows at different speeds for different colors of light.
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For example, the Fizeau experiment of 1851, conducted by French physicist Hippolyte Fizeau, demonstrated that the speed of light in flowing water was less than the sum of the speed of light in air plus the speed of the water by an amount dependent on the water's index of refraction.
Among other issues, the dependence of the partial aether-dragging implied by this experiment on the index of refraction (which is dependent on wavelength) led to the unpalatable conclusion that aether simultaneously flows at different speeds for different colors of light. The Michelson–Morley experiment of 1887 (Fig. 1-2) showed no differential influence of Earth's motions through the hypothetical aether on the speed of light, and the most likely explanation, complete aether dragging, was in conflict with the observation of stellar aberration.
George Francis FitzGerald in 1889, and Hendrik Lorentz in 1892, independently proposed that material bodies traveling through the fixed aether were physically affected by their passage, contracting in the direction of motion by an amount that was exactly what was necessary to explain the negative results of the Michelson–Morley experiment. No length changes occur in directions transverse to the direction of motion.
By 1904, Lorentz had expanded his theory such that he had arrived at equations formally identical with those that Einstein was to derive later, i.e. the Lorentz transformation.
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No length changes occur in directions transverse to the direction of motion.
By 1904, Lorentz had expanded his theory such that he had arrived at equations formally identical with those that Einstein was to derive later, i.e. the Lorentz transformation. As a theory of dynamics (the study of forces and torques and their effect on motion), his theory assumed actual physical deformations of the physical constituents of matter. Lorentz's equations predicted a quantity that he called local time, with which he could explain the aberration of light, the Fizeau experiment and other phenomena.
Henri Poincaré was the first to combine space and time into spacetime. He argued in 1898 that the simultaneity of two events is a matter of convention. In 1900, he recognized that Lorentz's "local time" is actually what is indicated by moving clocks by applying an explicitly operational definition of clock synchronization assuming constant light speed. In 1900 and 1904, he suggested the inherent undetectability of the aether by emphasizing the validity of what he called the principle of relativity. In 1905/1906 he mathematically perfected Lorentz's theory of electrons in order to bring it into accordance with the postulate of relativity.
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In 1900 and 1904, he suggested the inherent undetectability of the aether by emphasizing the validity of what he called the principle of relativity. In 1905/1906 he mathematically perfected Lorentz's theory of electrons in order to bring it into accordance with the postulate of relativity.
While discussing various hypotheses on Lorentz invariant gravitation, he introduced the innovative concept of a 4-dimensional spacetime by defining various four vectors, namely four-position, four-velocity, and four-force. He did not pursue the 4-dimensional formalism in subsequent papers, however, stating that this line of research seemed to "entail great pain for limited profit", ultimately concluding "that three-dimensional language seems the best suited to the description of our world". Even as late as 1909, Poincaré continued to describe the dynamical interpretation of the Lorentz transform.
In 1905, Albert Einstein analyzed special relativity in terms of kinematics (the study of moving bodies without reference to forces) rather than dynamics. His results were mathematically equivalent to those of Lorentz and Poincaré. He obtained them by recognizing that the entire theory can be built upon two postulates: the principle of relativity and the principle of the constancy of light speed.
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His results were mathematically equivalent to those of Lorentz and Poincaré. He obtained them by recognizing that the entire theory can be built upon two postulates: the principle of relativity and the principle of the constancy of light speed. His work was filled with vivid imagery involving the exchange of light signals between clocks in motion, careful measurements of the lengths of moving rods, and other such examples.
Einstein in 1905 superseded previous attempts of an electromagnetic mass–energy relation by introducing the general equivalence of mass and energy, which was instrumental for his subsequent formulation of the equivalence principle in 1907, which declares the equivalence of inertial and gravitational mass. By using the mass–energy equivalence, Einstein showed that the gravitational mass of a body is proportional to its energy content, which was one of the early results in developing general relativity. While it would appear that he did not at first think geometrically about spacetime, in the further development of general relativity, Einstein fully incorporated the spacetime formalism.
When Einstein published in 1905, another of his competitors, his former mathematics professor Hermann Minkowski, had also arrived at most of the basic elements of special relativity.
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While it would appear that he did not at first think geometrically about spacetime, in the further development of general relativity, Einstein fully incorporated the spacetime formalism.
When Einstein published in 1905, another of his competitors, his former mathematics professor Hermann Minkowski, had also arrived at most of the basic elements of special relativity. Max Born recounted a meeting he had made with Minkowski, seeking to be Minkowski's student/collaborator:
Minkowski had been concerned with the state of electrodynamics after Michelson's disruptive experiments at least since the summer of 1905, when Minkowski and David Hilbert led an advanced seminar attended by notable physicists of the time to study the papers of Lorentz, Poincaré et al. Minkowski saw Einstein's work as an extension of Lorentz's, and was most directly influenced by Poincaré.
On 5 November 1907 (a little more than a year before his death), Minkowski introduced his geometric interpretation of spacetime in a lecture to the Göttingen Mathematical society with the title, The Relativity Principle (Das Relativitätsprinzip). On 21 September 1908, Minkowski presented his talk, Space and Time (Raum und Zeit), to the German Society of Scientists and Physicians. The opening words of Space and Time include Minkowski's statement that "Henceforth, space for itself, and time for itself shall completely reduce to a mere shadow, and only some sort of union of the two shall preserve independence."
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On 21 September 1908, Minkowski presented his talk, Space and Time (Raum und Zeit), to the German Society of Scientists and Physicians. The opening words of Space and Time include Minkowski's statement that "Henceforth, space for itself, and time for itself shall completely reduce to a mere shadow, and only some sort of union of the two shall preserve independence." Space and Time included the first public presentation of spacetime diagrams (Fig. 1-4), and included a remarkable demonstration that the concept of the invariant interval (discussed below), along with the empirical observation that the speed of light is finite, allows derivation of the entirety of special relativity.{{refn|group=note|(In the following, the group G∞ is the Galilean group and the group Gc the Lorentz group.) "With respect to this it is clear that the group Gc in the limit for
The spacetime concept and the Lorentz group are closely connected to certain types of sphere, hyperbolic, or conformal geometries and their transformation groups already developed in the 19th century, in which invariant intervals analogous to the spacetime interval are used.
Einstein, for his part, was initially dismissive of Minkowski's geometric interpretation of special relativity, regarding it as überflüssige Gelehrsamkeit (superfluous learnedness). However, in order to complete his search for general relativity that started in 1907, the geometric interpretation of relativity proved to be vital.
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Einstein, for his part, was initially dismissive of Minkowski's geometric interpretation of special relativity, regarding it as überflüssige Gelehrsamkeit (superfluous learnedness). However, in order to complete his search for general relativity that started in 1907, the geometric interpretation of relativity proved to be vital. In 1916, Einstein fully acknowledged his indebtedness to Minkowski, whose interpretation greatly facilitated the transition to general relativity. Since there are other types of spacetime, such as the curved spacetime of general relativity, the spacetime of special relativity is today known as Minkowski spacetime.
## Spacetime in special relativity
### Spacetime interval
In three dimensions, the distance between two points can be defined using the Pythagorean theorem:
Although two viewers may measure the x, y, and z position of the two points using different coordinate systems, the distance between the points will be the same for both, assuming that they are measuring using the same units. The distance is "invariant".
In special relativity, however, the distance between two points is no longer the same if measured by two different observers, when one of the observers is moving, because of Lorentz contraction. The situation is even more complicated if the two points are separated in time as well as in space.
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In special relativity, however, the distance between two points is no longer the same if measured by two different observers, when one of the observers is moving, because of Lorentz contraction. The situation is even more complicated if the two points are separated in time as well as in space. For example, if one observer sees two events occur at the same place, but at different times, a person moving with respect to the first observer will see the two events occurring at different places, because the moving point of view sees itself as stationary, and the position of the event as receding or approaching. Thus, a different measure must be used to measure the effective "distance" between two events.
In four-dimensional spacetime, the analog to distance is the interval. Although time comes in as a fourth dimension, it is treated differently than the spatial dimensions. Minkowski space hence differs in important respects from four-dimensional Euclidean space. The fundamental reason for merging space and time into spacetime is that space and time are separately not invariant, which is to say that, under the proper conditions, different observers will disagree on the length of time between two events (because of time dilation) or the distance between the two events (because of length contraction). Special relativity provides a new invariant, called the spacetime interval, which combines distances in space and in time.
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The fundamental reason for merging space and time into spacetime is that space and time are separately not invariant, which is to say that, under the proper conditions, different observers will disagree on the length of time between two events (because of time dilation) or the distance between the two events (because of length contraction). Special relativity provides a new invariant, called the spacetime interval, which combines distances in space and in time. All observers who measure the time and distance between any two events will end up computing the same spacetime interval. Suppose an observer measures two events as being separated in time by and a spatial distance Then the squared spacetime interval between the two events that are separated by a distance in space and by in the -coordinate is: Extract of page 27
or for three space dimensions,
The constant the speed of light, converts time units (like seconds) into space units (like meters). The squared interval is a measure of separation between events A and B that are time separated and in addition space separated either because there are two separate objects undergoing events, or because a single object in space is moving inertially between its events. The separation interval is the difference between the square of the spatial distance separating event B from event A and the square of the spatial distance traveled by a light signal in that same time interval . If the event separation is due to a light signal, then this difference vanishes and .
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The separation interval is the difference between the square of the spatial distance separating event B from event A and the square of the spatial distance traveled by a light signal in that same time interval . If the event separation is due to a light signal, then this difference vanishes and .
When the event considered is infinitesimally close to each other, then we may write
In a different inertial frame, say with coordinates , the spacetime interval can be written in a same form as above. Because of the constancy of speed of light, the light events in all inertial frames belong to zero interval, . For any other infinitesimal event where , one can prove that
which in turn upon integration leads to .Landau, L. D., and Lifshitz, E. M. (2013). The classical theory of fields (Vol. 2). The invariance of the spacetime interval between the same events for all inertial frames of reference is one of the fundamental results of special theory of relativity.
Although for brevity, one frequently sees interval expressions expressed without deltas, including in most of the following discussion, it should be understood that in general, means , etc. We are always concerned with differences of spatial or temporal coordinate values belonging to two events, and since there is no preferred origin, single coordinate values have no essential meaning.
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Although for brevity, one frequently sees interval expressions expressed without deltas, including in most of the following discussion, it should be understood that in general, means , etc. We are always concerned with differences of spatial or temporal coordinate values belonging to two events, and since there is no preferred origin, single coordinate values have no essential meaning.
The equation above is similar to the Pythagorean theorem, except with a minus sign between the and the terms. The spacetime interval is the quantity not itself. The reason is that unlike distances in Euclidean geometry, intervals in Minkowski spacetime can be negative. Rather than deal with square roots of negative numbers, physicists customarily regard as a distinct symbol in itself, rather than the square of something.
Note: There are two sign conventions in use in the relativity literature:
and
These sign conventions are associated with the metric signatures and A minor variation is to place the time coordinate last rather than first. Both conventions are widely used within the field of study.
In the following discussion, we use the first convention.
In general can assume any real number value. If is positive, the spacetime interval is referred to as timelike. Since spatial distance traversed by any massive object is always less than distance traveled by the light for the same time interval, positive intervals are always timelike.
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If is positive, the spacetime interval is referred to as timelike. Since spatial distance traversed by any massive object is always less than distance traveled by the light for the same time interval, positive intervals are always timelike. If is negative, the spacetime interval is said to be spacelike. Spacetime intervals are equal to zero when In other words, the spacetime interval between two events on the world line of something moving at the speed of light is zero. Such an interval is termed lightlike or null. A photon arriving in our eye from a distant star will not have aged, despite having (from our perspective) spent years in its passage.
A spacetime diagram is typically drawn with only a single space and a single time coordinate. Fig. 2-1 presents a spacetime diagram illustrating the world lines (i.e. paths in spacetime) of two photons, A and B, originating from the same event and going in opposite directions. In addition, C illustrates the world line of a slower-than-light-speed object. The vertical time coordinate is scaled by so that it has the same units (meters) as the horizontal space coordinate. Since photons travel at the speed of light, their world lines have a slope of ±1. In other words, every meter that a photon travels to the left or right requires approximately 3.3 nanoseconds of time.
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Since photons travel at the speed of light, their world lines have a slope of ±1. In other words, every meter that a photon travels to the left or right requires approximately 3.3 nanoseconds of time.
Reference frames
To gain insight in how spacetime coordinates measured by observers in different reference frames compare with each other, it is useful to work with a simplified setup with frames in a standard configuration. With care, this allows simplification of the math with no loss of generality in the conclusions that are reached. In Fig. 2-2, two Galilean reference frames (i.e. conventional 3-space frames) are displayed in relative motion. Frame S belongs to a first observer O, and frame S′ (pronounced "S prime") belongs to a second observer O′.
- The x, y, z axes of frame S are oriented parallel to the respective primed axes of frame S′.
- Frame S′ moves in the x-direction of frame S with a constant velocity v as measured in frame S.
- The origins of frames S and S′ are coincident when time for frame S and for frame S′.
Fig. 2-3a redraws Fig. 2-2 in a different orientation. Fig. 2-3b illustrates a relativistic spacetime diagram from the viewpoint of observer O. Since S and S′ are in standard configuration, their origins coincide at times in frame S and in frame S′.
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Fig. 2-3b illustrates a relativistic spacetime diagram from the viewpoint of observer O. Since S and S′ are in standard configuration, their origins coincide at times in frame S and in frame S′. The axis passes through the events in frame S′ which have But the points with are moving in the x-direction of frame S with velocity v, so that they are not coincident with the ct axis at any time other than zero. Therefore, the axis is tilted with respect to the ct axis by an angle θ given by
The x′ axis is also tilted with respect to the x axis. To determine the angle of this tilt, we recall that the slope of the world line of a light pulse is always ±1. Fig. 2-3c presents a spacetime diagram from the viewpoint of observer O′. Event P represents the emission of a light pulse at The pulse is reflected from a mirror situated a distance a from the light source (event Q), and returns to the light source at (event R).
The same events P, Q, R are plotted in Fig. 2-3b in the frame of observer O. The light paths have and −1, so that △PQR forms a right triangle with PQ and QR both at 45 degrees to the x and ct axes. Since the angle between and must also be θ.
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The light paths have and −1, so that △PQR forms a right triangle with PQ and QR both at 45 degrees to the x and ct axes. Since the angle between and must also be θ.
While the rest frame has space and time axes that meet at right angles, the moving frame is drawn with axes that meet at an acute angle. The frames are actually equivalent. The asymmetry is due to unavoidable distortions in how spacetime coordinates can map onto a Cartesian plane, and should be considered no stranger than the manner in which, on a Mercator projection of the Earth, the relative sizes of land masses near the poles (Greenland and Antarctica) are highly exaggerated relative to land masses near the Equator.
### Light cone
In Fig. 2–4, event O is at the origin of a spacetime diagram, and the two diagonal lines represent all events that have zero spacetime interval with respect to the origin event. These two lines form what is called the light cone of the event O, since adding a second spatial dimension (Fig. 2-5) makes the appearance that of two right circular cones meeting with their apices at O. One cone extends into the future (t>0), the other into the past (t<0).
A light (double) cone divides spacetime into separate regions with respect to its apex.
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These two lines form what is called the light cone of the event O, since adding a second spatial dimension (Fig. 2-5) makes the appearance that of two right circular cones meeting with their apices at O. One cone extends into the future (t>0), the other into the past (t<0).
A light (double) cone divides spacetime into separate regions with respect to its apex. The interior of the future light cone consists of all events that are separated from the apex by more time (temporal distance) than necessary to cross their spatial distance at lightspeed; these events comprise the timelike future of the event O. Likewise, the timelike past comprises the interior events of the past light cone. So in timelike intervals Δct is greater than Δx, making timelike intervals positive.
The region exterior to the light cone consists of events that are separated from the event O by more space than can be crossed at lightspeed in the given time. These events comprise the so-called spacelike region of the event O, denoted "Elsewhere" in Fig. 2-4. Events on the light cone itself are said to be lightlike (or null separated) from O. Because of the invariance of the spacetime interval, all observers will assign the same light cone to any given event, and thus will agree on this division of spacetime.
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These events comprise the so-called spacelike region of the event O, denoted "Elsewhere" in Fig. 2-4. Events on the light cone itself are said to be lightlike (or null separated) from O. Because of the invariance of the spacetime interval, all observers will assign the same light cone to any given event, and thus will agree on this division of spacetime.
The light cone has an essential role within the concept of causality. It is possible for a not-faster-than-light-speed signal to travel from the position and time of O to the position and time of D (Fig. 2-4). It is hence possible for event O to have a causal influence on event D. The future light cone contains all the events that could be causally influenced by O. Likewise, it is possible for a not-faster-than-light-speed signal to travel from the position and time of A, to the position and time of O. The past light cone contains all the events that could have a causal influence on O. In contrast, assuming that signals cannot travel faster than the speed of light, any event, like e.g. B or C, in the spacelike region (Elsewhere), cannot either affect event O, nor can they be affected by event O employing such signalling. Under this assumption any causal relationship between event O and any events in the spacelike region of a light cone is excluded.
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In contrast, assuming that signals cannot travel faster than the speed of light, any event, like e.g. B or C, in the spacelike region (Elsewhere), cannot either affect event O, nor can they be affected by event O employing such signalling. Under this assumption any causal relationship between event O and any events in the spacelike region of a light cone is excluded.
### Relativity of simultaneity
All observers will agree that for any given event, an event within the given event's future light cone occurs after the given event. Likewise, for any given event, an event within the given event's past light cone occurs before the given event. The before–after relationship observed for timelike-separated events remains unchanged no matter what the reference frame of the observer, i.e. no matter how the observer may be moving. The situation is quite different for spacelike-separated events. Fig. 2-4 was drawn from the reference frame of an observer moving at From this reference frame, event C is observed to occur after event O, and event B is observed to occur before event O.
From a different reference frame, the orderings of these non-causally-related events can be reversed. In particular, one notes that if two events are simultaneous in a particular reference frame, they are necessarily separated by a spacelike interval and thus are noncausally related.
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Fig. 2-4 was drawn from the reference frame of an observer moving at From this reference frame, event C is observed to occur after event O, and event B is observed to occur before event O.
From a different reference frame, the orderings of these non-causally-related events can be reversed. In particular, one notes that if two events are simultaneous in a particular reference frame, they are necessarily separated by a spacelike interval and thus are noncausally related. The observation that simultaneity is not absolute, but depends on the observer's reference frame, is termed the relativity of simultaneity.
Fig. 2-6 illustrates the use of spacetime diagrams in the analysis of the relativity of simultaneity. The events in spacetime are invariant, but the coordinate frames transform as discussed above for Fig. 2-3. The three events are simultaneous from the reference frame of an observer moving at From the reference frame of an observer moving at the events appear to occur in the order From the reference frame of an observer moving at , the events appear to occur in the order . The white line represents a plane of simultaneity being moved from the past of the observer to the future of the observer, highlighting events residing on it. The gray area is the light cone of the observer, which remains invariant.
A spacelike spacetime interval gives the same distance that an observer would measure if the events being measured were simultaneous to the observer.
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The gray area is the light cone of the observer, which remains invariant.
A spacelike spacetime interval gives the same distance that an observer would measure if the events being measured were simultaneous to the observer. A spacelike spacetime interval hence provides a measure of proper distance, i.e. the true distance = Likewise, a timelike spacetime interval gives the same measure of time as would be presented by the cumulative ticking of a clock that moves along a given world line. A timelike spacetime interval hence provides a measure of the proper time =
### Invariant hyperbola
In Euclidean space (having spatial dimensions only), the set of points equidistant (using the Euclidean metric) from some point form a circle (in two dimensions) or a sphere (in three dimensions). In Minkowski spacetime (having one temporal and one spatial dimension), the points at some constant spacetime interval away from the origin (using the Minkowski metric) form curves given by the two equations
with some positive real constant. These equations describe two families of hyperbolae in an x–ct spacetime diagram, which are termed invariant hyperbolae.
In Fig. 2-7a, each magenta hyperbola connects all events having some fixed spacelike separation from the origin, while the green hyperbolae connect events of equal timelike separation.
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These equations describe two families of hyperbolae in an x–ct spacetime diagram, which are termed invariant hyperbolae.
In Fig. 2-7a, each magenta hyperbola connects all events having some fixed spacelike separation from the origin, while the green hyperbolae connect events of equal timelike separation.
The magenta hyperbolae, which cross the x axis, are timelike curves, which is to say that these hyperbolae represent actual paths that can be traversed by (constantly accelerating) particles in spacetime: Between any two events on one hyperbola a causality relation is possible, because the inverse of the slope—representing the necessary speed—for all secants is less than . On the other hand, the green hyperbolae, which cross the ct axis, are spacelike curves because all intervals along these hyperbolae are spacelike intervals: No causality is possible between any two points on one of these hyperbolae, because all secants represent speeds larger than .
Fig. 2-7b reflects the situation in Minkowski spacetime (one temporal and two spatial dimensions) with the corresponding hyperboloids. The invariant hyperbolae displaced by spacelike intervals from the origin generate hyperboloids of one sheet, while the invariant hyperbolae displaced by timelike intervals from the origin generate hyperboloids of two sheets.
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Fig. 2-7b reflects the situation in Minkowski spacetime (one temporal and two spatial dimensions) with the corresponding hyperboloids. The invariant hyperbolae displaced by spacelike intervals from the origin generate hyperboloids of one sheet, while the invariant hyperbolae displaced by timelike intervals from the origin generate hyperboloids of two sheets.
The (1+2)-dimensional boundary between space- and time-like hyperboloids, established by the events forming a zero spacetime interval to the origin, is made up by degenerating the hyperboloids to the light cone. In (1+1)-dimensions the hyperbolae degenerate to the two grey 45°-lines depicted in Fig. 2-7a.
### Time dilation and length contraction
Fig. 2-8 illustrates the invariant hyperbola for all events that can be reached from the origin in a proper time of 5 meters (approximately ). Different world lines represent clocks moving at different speeds. A clock that is stationary with respect to the observer has a world line that is vertical, and the elapsed time measured by the observer is the same as the proper time. For a clock traveling at 0.3 c, the elapsed time measured by the observer is 5.24 meters (), while for a clock traveling at 0.7 c, the elapsed time measured by the observer is 7.00 meters ().
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A clock that is stationary with respect to the observer has a world line that is vertical, and the elapsed time measured by the observer is the same as the proper time. For a clock traveling at 0.3 c, the elapsed time measured by the observer is 5.24 meters (), while for a clock traveling at 0.7 c, the elapsed time measured by the observer is 7.00 meters ().
This illustrates the phenomenon known as time dilation. Clocks that travel faster take longer (in the observer frame) to tick out the same amount of proper time, and they travel further along the x–axis within that proper time than they would have without time dilation. The measurement of time dilation by two observers in different inertial reference frames is mutual. If observer O measures the clocks of observer O′ as running slower in his frame, observer O′ in turn will measure the clocks of observer O as running slower.
Length contraction, like time dilation, is a manifestation of the relativity of simultaneity. Measurement of length requires measurement of the spacetime interval between two events that are simultaneous in one's frame of reference. But events that are simultaneous in one frame of reference are, in general, not simultaneous in other frames of reference.
Fig. 2-9 illustrates the motions of a 1 m rod that is traveling at 0.5 c along the x axis.
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But events that are simultaneous in one frame of reference are, in general, not simultaneous in other frames of reference.
Fig. 2-9 illustrates the motions of a 1 m rod that is traveling at 0.5 c along the x axis. The edges of the blue band represent the world lines of the rod's two endpoints. The invariant hyperbola illustrates events separated from the origin by a spacelike interval of 1 m. The endpoints O and B measured when = 0 are simultaneous events in the S′ frame. But to an observer in frame S, events O and B are not simultaneous. To measure length, the observer in frame S measures the endpoints of the rod as projected onto the x-axis along their world lines. The projection of the rod's world sheet onto the x axis yields the foreshortened length OC.
(not illustrated) Drawing a vertical line through A so that it intersects the x′ axis demonstrates that, even as OB is foreshortened from the point of view of observer O, OA is likewise foreshortened from the point of view of observer O′. In the same way that each observer measures the other's clocks as running slow, each observer measures the other's rulers as being contracted.
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The projection of the rod's world sheet onto the x axis yields the foreshortened length OC.
(not illustrated) Drawing a vertical line through A so that it intersects the x′ axis demonstrates that, even as OB is foreshortened from the point of view of observer O, OA is likewise foreshortened from the point of view of observer O′. In the same way that each observer measures the other's clocks as running slow, each observer measures the other's rulers as being contracted.
In regards to mutual length contraction, Fig. 2-9 illustrates that the primed and unprimed frames are mutually rotated by a hyperbolic angle (analogous to ordinary angles in Euclidean geometry).In a Cartesian plane, ordinary rotation leaves a circle unchanged. In spacetime, hyperbolic rotation preserves the hyperbolic metric. Because of this rotation, the projection of a primed meter-stick onto the unprimed x-axis is foreshortened, while the projection of an unprimed meter-stick onto the primed x′-axis is likewise foreshortened.
###
#### Mutual time dilation
and the twin paradox
Mutual time dilation
Mutual time dilation and length contraction tend to strike beginners as inherently self-contradictory concepts.
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Because of this rotation, the projection of a primed meter-stick onto the unprimed x-axis is foreshortened, while the projection of an unprimed meter-stick onto the primed x′-axis is likewise foreshortened.
###
#### Mutual time dilation
and the twin paradox
Mutual time dilation
Mutual time dilation and length contraction tend to strike beginners as inherently self-contradictory concepts. If an observer in frame S measures a clock, at rest in frame S', as running slower than his', while S' is moving at speed v in S, then the principle of relativity requires that an observer in frame S' likewise measures a clock in frame S, moving at speed −v in S', as running slower than hers. How two clocks can run both slower than the other, is an important question that "goes to the heart of understanding special relativity. "
This apparent contradiction stems from not correctly taking into account the different settings of the necessary, related measurements. These settings allow for a consistent explanation of the only apparent contradiction. It is not about the abstract ticking of two identical clocks, but about how to measure in one frame the temporal distance of two ticks of a moving clock. It turns out that in mutually observing the duration between ticks of clocks, each moving in the respective frame, different sets of clocks must be involved.
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It is not about the abstract ticking of two identical clocks, but about how to measure in one frame the temporal distance of two ticks of a moving clock. It turns out that in mutually observing the duration between ticks of clocks, each moving in the respective frame, different sets of clocks must be involved. In order to measure in frame S the tick duration of a moving clock W′ (at rest in S′), one uses two additional, synchronized clocks W1 and W2 at rest in two arbitrarily fixed points in S with the spatial distance d.
Two events can be defined by the condition "two clocks are simultaneously at one place", i.e., when W′ passes each W1 and W2. For both events the two readings of the collocated clocks are recorded. The difference of the two readings of W1 and W2 is the temporal distance of the two events in S, and their spatial distance is d.
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For both events the two readings of the collocated clocks are recorded. The difference of the two readings of W1 and W2 is the temporal distance of the two events in S, and their spatial distance is d. The difference of the two readings of W′ is the temporal distance of the two events in S′. In S′ these events are only separated in time, they happen at the same place in S′. Because of the invariance of the spacetime interval spanned by these two events, and the nonzero spatial separation d in S, the temporal distance in S′ must be smaller than the one in S: the smaller temporal distance between the two events, resulting from the readings of the moving clock W′, belongs to the slower running clock W′.
Conversely, for judging in frame S′ the temporal distance of two events on a moving clock W (at rest in S), one needs two clocks at rest in S′.
In this comparison the clock W is moving by with velocity −v.
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The difference of the two readings of W′ is the temporal distance of the two events in S′. In S′ these events are only separated in time, they happen at the same place in S′. Because of the invariance of the spacetime interval spanned by these two events, and the nonzero spatial separation d in S, the temporal distance in S′ must be smaller than the one in S: the smaller temporal distance between the two events, resulting from the readings of the moving clock W′, belongs to the slower running clock W′.
Conversely, for judging in frame S′ the temporal distance of two events on a moving clock W (at rest in S), one needs two clocks at rest in S′.
In this comparison the clock W is moving by with velocity −v. Recording again the four readings for the events, defined by "two clocks simultaneously at one place", results in the analogous temporal distances of the two events, now temporally and spatially separated in S′, and only temporally separated but collocated in S. To keep the spacetime interval invariant, the temporal distance in S must be smaller than in S′, because of the spatial separation of the events in S′: now clock W is observed to run slower.
The necessary recordings for the two judgements, with "one moving clock" and "two clocks at rest" in respectively S or S′, involves two different sets, each with three clocks.
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Recording again the four readings for the events, defined by "two clocks simultaneously at one place", results in the analogous temporal distances of the two events, now temporally and spatially separated in S′, and only temporally separated but collocated in S. To keep the spacetime interval invariant, the temporal distance in S must be smaller than in S′, because of the spatial separation of the events in S′: now clock W is observed to run slower.
The necessary recordings for the two judgements, with "one moving clock" and "two clocks at rest" in respectively S or S′, involves two different sets, each with three clocks. Since there are different sets of clocks involved in the measurements, there is no inherent necessity that the measurements be reciprocally "consistent" such that, if one observer measures the moving clock to be slow, the other observer measures the one's clock to be fast.
Fig. 2-10 illustrates the previous discussion of mutual time dilation with Minkowski diagrams.
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Fig. 2-10 illustrates the previous discussion of mutual time dilation with Minkowski diagrams. The upper picture reflects the measurements as seen from frame S "at rest" with unprimed, rectangular axes, and frame S′ "moving with v > 0", coordinatized by primed, oblique axes, slanted to the right; the lower picture shows frame S′ "at rest" with primed, rectangular coordinates, and frame S "moving with −v < 0", with unprimed, oblique axes, slanted to the left.
Each line drawn parallel to a spatial axis (x, x′) represents a line of simultaneity. All events on such a line have the same time value (ct, ct′). Likewise, each line drawn parallel to a temporal axis (ct, ct′) represents a line of equal spatial coordinate values (x, x′).
One may designate in both pictures the origin O (= ) as the event, where the respective "moving clock" is collocated with the "first clock at rest" in both comparisons. Obviously, for this event the readings on both clocks in both comparisons are zero. As a consequence, the worldlines of the moving clocks are the slanted to the right ct′-axis (upper pictures, clock W′) and the slanted to the left ct-axes (lower pictures, clock W).
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Obviously, for this event the readings on both clocks in both comparisons are zero. As a consequence, the worldlines of the moving clocks are the slanted to the right ct′-axis (upper pictures, clock W′) and the slanted to the left ct-axes (lower pictures, clock W). The worldlines of W1 and W′1 are the corresponding vertical time axes (ct in the upper pictures, and ct′ in the lower pictures).
In the upper picture the place for W2 is taken to be Ax > 0, and thus the worldline (not shown in the pictures) of this clock intersects the worldline of the moving clock (the ct′-axis) in the event labelled A, where "two clocks are simultaneously at one place". In the lower picture the place for W′2 is taken to be Cx′ < 0, and so in this measurement the moving clock W passes W′2 in the event C.
In the upper picture the ct-coordinate At of the event A (the reading of W2) is labeled B, thus giving the elapsed time between the two events, measured with W1 and W2, as OB. For a comparison, the length of the time interval OA, measured with W′, must be transformed to the scale of the ct-axis. This is done by the invariant hyperbola (see also Fig.
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For a comparison, the length of the time interval OA, measured with W′, must be transformed to the scale of the ct-axis. This is done by the invariant hyperbola (see also Fig. 2-8) through A, connecting all events with the same spacetime interval from the origin as A. This yields the event C on the ct-axis, and obviously: OC < OB, the "moving" clock W′ runs slower.
To show the mutual time dilation immediately in the upper picture, the event D may be constructed as the event at x′ = 0 (the location of clock W′ in S′), that is simultaneous to C (OC has equal spacetime interval as OA) in S′. This shows that the time interval OD is longer than OA, showing that the "moving" clock runs slower.
In the lower picture the frame S is moving with velocity −v in the frame S′ at rest. The worldline of clock W is the ct-axis (slanted to the left), the worldline of W′1 is the vertical ct′-axis, and the worldline of W′2 is the vertical through event C, with ct′-coordinate D. The invariant hyperbola through event C scales the time interval OC to OA, which is shorter than OD; also, B is constructed (similar to D in the upper pictures) as simultaneous to A in S, at x = 0.
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The worldline of clock W is the ct-axis (slanted to the left), the worldline of W′1 is the vertical ct′-axis, and the worldline of W′2 is the vertical through event C, with ct′-coordinate D. The invariant hyperbola through event C scales the time interval OC to OA, which is shorter than OD; also, B is constructed (similar to D in the upper pictures) as simultaneous to A in S, at x = 0. The result OB > OC corresponds again to above.
The word "measure" is important. In classical physics an observer cannot affect an observed object, but the object's state of motion can affect the observer's observations of the object.
#### Twin paradox
Many introductions to special relativity illustrate the differences between Galilean relativity and special relativity by posing a series of "paradoxes". These paradoxes are, in fact, ill-posed problems, resulting from our unfamiliarity with velocities comparable to the speed of light. The remedy is to solve many problems in special relativity and to become familiar with its so-called counter-intuitive predictions. The geometrical approach to studying spacetime is considered one of the best methods for developing a modern intuition.
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The remedy is to solve many problems in special relativity and to become familiar with its so-called counter-intuitive predictions. The geometrical approach to studying spacetime is considered one of the best methods for developing a modern intuition.
The twin paradox is a thought experiment involving identical twins, one of whom makes a journey into space in a high-speed rocket, returning home to find that the twin who remained on Earth has aged more. This result appears puzzling because each twin observes the other twin as moving, and so at first glance, it would appear that each should find the other to have aged less. The twin paradox sidesteps the justification for mutual time dilation presented above by avoiding the requirement for a third clock. Nevertheless, the twin paradox is not a true paradox because it is easily understood within the context of special relativity.
The impression that a paradox exists stems from a misunderstanding of what special relativity states. Special relativity does not declare all frames of reference to be equivalent, only inertial frames. The traveling twin's frame is not inertial during periods when she is accelerating. Furthermore, the difference between the twins is observationally detectable: the traveling twin needs to fire her rockets to be able to return home, while the stay-at-home twin does not.
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The traveling twin's frame is not inertial during periods when she is accelerating. Furthermore, the difference between the twins is observationally detectable: the traveling twin needs to fire her rockets to be able to return home, while the stay-at-home twin does not. Even with no (de)acceleration i.e. using one inertial frame O for constant, high-velocity outward journey and another inertial frame I for constant, high-velocity inward journey – the sum of the elapsed time in those frames (O and I) is shorter than the elapsed time in the stationary inertial frame S. Thus acceleration and deceleration is not the cause of shorter elapsed time during the outward and inward journey. Instead the use of two different constant, high-velocity inertial frames for outward and inward journey is really the cause of shorter elapsed time total. Granted, if the same twin has to travel outward and inward leg of the journey and safely switch from outward to inward leg of the journey, the acceleration and deceleration is required. If the travelling twin could ride the high-velocity outward inertial frame and instantaneously switch to high-velocity inward inertial frame the example would still work. The point is that real reason should be stated clearly.
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If the travelling twin could ride the high-velocity outward inertial frame and instantaneously switch to high-velocity inward inertial frame the example would still work. The point is that real reason should be stated clearly. The asymmetry is because of the comparison of sum of elapsed times in two different inertial frames (O and I) to the elapsed time in a single inertial frame S.
These distinctions should result in a difference in the twins' ages. The spacetime diagram of Fig. 2-11 presents the simple case of a twin going straight out along the x axis and immediately turning back. From the standpoint of the stay-at-home twin, there is nothing puzzling about the twin paradox at all. The proper time measured along the traveling twin's world line from O to C, plus the proper time measured from C to B, is less than the stay-at-home twin's proper time measured from O to A to B. More complex trajectories require integrating the proper time between the respective events along the curve (i.e. the path integral) to calculate the total amount of proper time experienced by the traveling twin.
Complications arise if the twin paradox is analyzed from the traveling twin's point of view.
Weiss's nomenclature, designating the stay-at-home twin as Terence and the traveling twin as Stella, is hereafter used.
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Complications arise if the twin paradox is analyzed from the traveling twin's point of view.
Weiss's nomenclature, designating the stay-at-home twin as Terence and the traveling twin as Stella, is hereafter used.
Stella is not in an inertial frame. Given this fact, it is sometimes incorrectly stated that full resolution of the twin paradox requires general relativity:
Although general relativity is not required to analyze the twin paradox, application of the Equivalence Principle of general relativity does provide some additional insight into the subject. Stella is not stationary in an inertial frame. Analyzed in Stella's rest frame, she is motionless for the entire trip. When she is coasting her rest frame is inertial, and Terence's clock will appear to run slow. But when she fires her rockets for the turnaround, her rest frame is an accelerated frame and she experiences a force which is pushing her as if she were in a gravitational field. Terence will appear to be high up in that field and because of gravitational time dilation, his clock will appear to run fast, so much so that the net result will be that Terence has aged more than Stella when they are back together. The theoretical arguments predicting gravitational time dilation are not exclusive to general relativity.
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Terence will appear to be high up in that field and because of gravitational time dilation, his clock will appear to run fast, so much so that the net result will be that Terence has aged more than Stella when they are back together. The theoretical arguments predicting gravitational time dilation are not exclusive to general relativity. Any theory of gravity will predict gravitational time dilation if it respects the principle of equivalence, including Newton's theory.
### Gravitation
This introductory section has focused on the spacetime of special relativity, since it is the easiest to describe. Minkowski spacetime is flat, takes no account of gravity, is uniform throughout, and serves as nothing more than a static background for the events that take place in it. The presence of gravity greatly complicates the description of spacetime. In general relativity, spacetime is no longer a static background, but actively interacts with the physical systems that it contains. Spacetime curves in the presence of matter, can propagate waves, bends light, and exhibits a host of other phenomena. A few of these phenomena are described in the later sections of this article.
## Basic mathematics of spacetime
### Galilean transformations
A basic goal is to be able to compare measurements made by observers in relative motion.
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## Basic mathematics of spacetime
### Galilean transformations
A basic goal is to be able to compare measurements made by observers in relative motion. If there is an observer O in frame S who has measured the time and space coordinates of an event, assigning this event three Cartesian coordinates and the time as measured on his lattice of synchronized clocks (see Fig. 1-1). A second observer O′ in a different frame S′ measures the same event in her coordinate system and her lattice of synchronized clocks . With inertial frames, neither observer is under acceleration, and a simple set of equations allows us to relate coordinates to . Given that the two coordinate systems are in standard configuration, meaning that they are aligned with parallel coordinates and that when , the coordinate transformation is as follows:
Fig. 3-1 illustrates that in Newton's theory, time is universal, not the velocity of light. Consider the following thought experiment: The red arrow illustrates a train that is moving at 0.4 c with respect to the platform. Within the train, a passenger shoots a bullet with a speed of 0.4 c in the frame of the train. The blue arrow illustrates that a person standing on the train tracks measures the bullet as traveling at 0.8 c. This is in accordance with our naive expectations.
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The blue arrow illustrates that a person standing on the train tracks measures the bullet as traveling at 0.8 c. This is in accordance with our naive expectations.
More generally, assuming that frame S′ is moving at velocity v with respect to frame S, then within frame S′, observer O′ measures an object moving with velocity . Velocity u with respect to frame S, since , , and , can be written as = = . This leads to and ultimately
or
which is the common-sense Galilean law for the addition of velocities.
### Relativistic composition of velocities
The composition of velocities is quite different in relativistic spacetime. To reduce the complexity of the equations slightly, we introduce a common shorthand for the ratio of the speed of an object relative to light,
Fig. 3-2a illustrates a red train that is moving forward at a speed given by . From the primed frame of the train, a passenger shoots a bullet with a speed given by , where the distance is measured along a line parallel to the red axis rather than parallel to the black x axis. What is the composite velocity u of the bullet relative to the platform, as represented by the blue arrow? Referring to Fig. 3-2b:
1. From the platform, the composite speed of the bullet is given by .
1.
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From the platform, the composite speed of the bullet is given by .
1. The two yellow triangles are similar because they are right triangles that share a common angle α. In the large yellow triangle, the ratio .
1. The ratios of corresponding sides of the two yellow triangles are constant, so that = . So and .
1. Substitute the expressions for b and r into the expression for u in step 1 to yield Einstein's formula for the addition of velocities:
1. :
The relativistic formula for addition of velocities presented above exhibits several important features:
- If and v are both very small compared with the speed of light, then the product /c2 becomes vanishingly small, and the overall result becomes indistinguishable from the Galilean formula (Newton's formula) for the addition of velocities: u = + v. The Galilean formula is a special case of the relativistic formula applicable to low velocities.
- If is set equal to c, then the formula yields u = c regardless of the starting value of v. The velocity of light is the same for all observers regardless their motions relative to the emitting source.
### Time dilation and length contraction revisited
It is straightforward to obtain quantitative expressions for time dilation and length contraction. Fig.
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### Time dilation and length contraction revisited
It is straightforward to obtain quantitative expressions for time dilation and length contraction. Fig. 3-3 is a composite image containing individual frames taken from two previous animations, simplified and relabeled for the purposes of this section.
To reduce the complexity of the equations slightly, there are a variety of different shorthand notations for ct:
and are common.
One also sees very frequently the use of the convention
In Fig. 3-3a, segments OA and OK represent equal spacetime intervals. Time dilation is represented by the ratio OB/OK. The invariant hyperbola has the equation where k = OK, and the red line representing the world line of a particle in motion has the equation w = x/β = xc/v. A bit of algebraic manipulation yields
The expression involving the square root symbol appears very frequently in relativity, and one over the expression is called the Lorentz factor, denoted by the Greek letter gamma :
If v is greater than or equal to c, the expression for becomes physically meaningless, implying that c is the maximum possible speed in nature. For any v greater than zero, the Lorentz factor will be greater than one, although the shape of the curve is such that for low speeds, the Lorentz factor is extremely close to one.
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If v is greater than or equal to c, the expression for becomes physically meaningless, implying that c is the maximum possible speed in nature. For any v greater than zero, the Lorentz factor will be greater than one, although the shape of the curve is such that for low speeds, the Lorentz factor is extremely close to one.
In Fig. 3-3b, segments OA and OK represent equal spacetime intervals. Length contraction is represented by the ratio OB/OK. The invariant hyperbola has the equation , where k = OK, and the edges of the blue band representing the world lines of the endpoints of a rod in motion have slope 1/β = c/v. Event A has coordinates
(x, w) = (γk, γβk). Since the tangent line through A and B has the equation w = (x − OB)/β, we have γβk = (γk − OB)/β and
### Lorentz transformations
The Galilean transformations and their consequent commonsense law of addition of velocities work well in our ordinary low-speed world of planes, cars and balls. Beginning in the mid-1800s, however, sensitive scientific instrumentation began finding anomalies that did not fit well with the ordinary addition of velocities.
Lorentz transformations are used to transform the coordinates of an event from one frame to another in special relativity.
The Lorentz factor appears in the Lorentz transformations:
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