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What we need to do is to get the students to understand that they can go out and do things that are not in their own hands.
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Hello, you're listening to The Science of Everything podcast, episode 145.
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Relativity and black Holes.
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I'm your host, James Fodor.
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This episode is a continuation of the discussion of general relativity, which we began in episode...
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136, which is the prerequisite for this episode.
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In that episode, we talked about general relativity and explained the notion of space-time, and how we describe velocity, distance and curvature of space-time using mathematical formalisms.
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And how we combine these formalisms together.
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To yield Einstein's field equations, which loosely say that, the curvature of space-time.
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Is proportional to the energy and matter content of spacetime.
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And I explained how Einstein's field equations are a series of 10.
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Coupled nonlinear partial differential equations.
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Which means that they're very complex and difficult to solve.
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For any realistic cases.
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However, I did say that there are some closed form, meaning sort of simply mathematically describable.
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Solutions known.
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To Einstein's field equations, and I'll talk about them in a future episode.
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Well, now is that future episode, or.
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One of those future episodes where we'll talk about solutions to Einstein's field equations, and in particular, in this episode, we're going to focus on the Schwarzschild metric.
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And how it's able to describe Well.
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Described the existence of black holes.
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So we'll talk about deriving the Schwarzschild metric, how to interpret the resulting metric.
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And then we'll see how.
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Z resulting metric.
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Yields predictions which have been experimentally verified and thereby serving as experimental evidence in favour of general relativity.
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We'll then talk in more detail about Schwarzschild black holes.
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Some of the phenomena there, like the event horizon, singularity and so forth.
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And we'll conclude by discussing some of the unsolved problems or outstanding issues with black holes, including the phenomena of hawking radiation, the no-hair theorem, and the black hole information Paradox.
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This will be a pretty dense episode, so hope you're ready.
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And let's then jump into it.
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But bear in mind, though, I will be assuming...
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That you've listened to.
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On general relativity, because that introduces some of the key ideas that I'll...
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We'll start with exactly where we picked up last time, which is Einstein's Field equations, a series of 10, coupled, nonlinear, partial differential equations relating the curvature of space and time to the energy and matter content of space and time.
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What we're going to do is try to find a solution to these equations.
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There are a number of closed form solutions known.
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We're going to focus on one of them today.
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And essentially, what this amounts to is solving for the equations to find the metric.
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Mathematical description of the overall shape of space and time.
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A metric that satisfies the equations.
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So the term on the left of the equations is the Einstein tensor.
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Denoted as a capital, G.
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And it more or less describes the curvature of the metric.
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Which represents the structure of space and time.
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On the right hand side is the stress energy tensile, which describes the energy content of of space and time.
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There's sort of two ways to solve.
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These couple of equations.
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One is to postulate a stress energy tensor, so to stipulate what the energy content of space is, and then solve for the metric.
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Given далее energy content.
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You can start with the metric.
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So specify what, though.
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And then solve for the stress energy tensor that will give you that metric.
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So in this particular case, the way we're going to do it, is we're going to stipulate what the stress-energy tensor is.
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As well as making a few other assumptions.
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And then we're going to see what metric that gives us, what metric satisfies.
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The equations when we stipulate what the energy content of the universe is.
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So we're going to stipulate at the outset that the energy and matter content of the universe, or at least the region of the universe that we're considering.
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Is going to be zero.
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So we're looking for a vacuum solution, it's often called, an empty space solution of the equation.
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So obviously that simplifies things dramatically.
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Because the right hand side of the equation is just zero.
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And it the equation, simplified down to the the rishi tends, is equal to zero.
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Remember, basically describes the curvature.
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Of um space of time in that region.
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So the next step, now that we've already simplified things quite a lot, is to make further assumptions to help simplify things.
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The next thing that we're going to assume is that the Rishi Tensor, or all of the components of the Rishi Tensor, are independent of time, so they're static.
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They're constant over time.
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So this is just representing the type of solution that we're looking for, which is a static solution.
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Again, this is often done for simplicity.
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Now it turns out, when you make this assumption, this dramatically simplifies things further.
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Because it means that any terms that interact with the time coordinate, Remember, there's four coordinates.
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Of, of space and time.
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There's one time coordinate and three spatial coordinates.
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So any of them that interact with the time coordinate have to go to zero.
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There will be changes over time.
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Making this assumption of a static field means that instead of having 16 components, of the Rishi Tensor, now there are only going to be four components.
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And they're just the diagonal components.
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So those along the diagonal.
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A 4x4 matrix is only the ones along the diagonal.
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That will be non-zero, everything else goes to zero.
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When we make this assumption of a static field.
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Things are now very simple because instead of having these 10...
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Coupled partial, different, nonlinear, partial, differential equations.
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Now we've got, we've reduced it down to four much simpler equations.
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All of which are just equal to zero.
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Four equations for each, essentially one for each of the coordinates, a one-time equation.
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And one for each of the three spatial coordinates.
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Now, to make even more simplifications, we introduce a third assumption, which is that we're going to look for a spherically symmetrical solution.
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So we're kind of interested in solutions that look the same, when you rotate them.
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That simplifies things a lot further because now we only have to worry essentially about two Coordinates.
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The radius, which is the distance from the center.
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And then one angle coordinate.
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Even with all of these simplifications, the equations that we have to solve are still somewhat complicated.
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But the algebra is at least now solvable, and obviously I can't go through all the details here.
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But at this point, what we need to do is take the highly simplified form of the Rishi tensor that we've derived by making these assumptions.
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And then just substitute in for the actual form of the Rishitensa.
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Remember, the Rishitensa is defined in terms of mathematical objects called Christoffel symbols.
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Basically, describe the way in which our path changes, owing to the curvature of space as we move, remember, in the last episode, we talked about the idea of someone holding a spear out in front of them?
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And starting at the North Pole and walking down towards the equator.
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Their spear will change direction as they walk along, even if they, it doesn't.
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