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Locally, doesn't look like they're tilting their spear at all, they're holding it in front of them.
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But it will actually change direction globally simply because the Earth is curving under them.
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And so, if you're to describe how the direction of that spear is changing with the motion of the the the person walking with it, you need to consider not just whether...
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Person is rotating or tilting their spear, but also the change in direction of the spear.
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Because of the curvature of the earth itself, the path that they're traveling on.
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Christoffel symbols help us to do that.
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That helps us to describe the change in direction of a path over a curved geometry.
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And so the Rishi tensor is defined in terms of these Christoffel symbols.
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Christoffel symbols, in turn, are defined in terms of various derivatives of the metric.
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The metric is the 4x4 array.
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That describes the overall shape.
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What we're going to do at this point with we've got the general form with the Rishi Tensor.
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In order to solve for the actual you final equation here, all we need to do is plug in.
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The definition of the different components of the Rishi Tensa in terms of Christoffelson was immense in terms of the metric.
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And then equate that all to zero, because, remember, each of these components is equal to zero.
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There's four components of the Rishi tensor, the diagonal components, one for each of the space-time coordinates, they're all equal to zero.
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Because of the simplifying assumption that we made of a vacuum solution.
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So we're just going to substitute in the correct forms for each of these components.
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And equate them all to zero.
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And we've already simplified the equations a lot because of the...
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Additional assumptions of the static field.
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And spherical symmetry that we've assumed.
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We substitute in the form of the Rishitensa.
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Do the TDC algebra, rearrange and combine some things together.
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And then once we're done, we end up with a solution for the metric.
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And this metric is called the Schwarzschild Metric after the...
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Scientist who derived it originally.
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So the Schwarzschild metric...
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Is, it turns out...
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Not just a solution, to Einstein's field equations.
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It's actually that there's a theorem called Burkhoff's theorem.
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But there's a theorem which states that the Schwarzschild metric is the only spherically symmetric vacuum solution of Einstein's field equations.
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So there's only one solution to Einstein's field equations.
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That is spherically symmetric.
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Describing a vacuum so no matter or energy.
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So that's quite an important result, and it turns out that it's not that difficult to derive.
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Closed form solutions of Einstein's field equations, but this is...
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Really, the simplest one.
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Um, of interest.
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And so there's been a lot of, well...
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Over the decades, there's been a lot of study into this metric and what it tells us.
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And it turns out that it actually describes a lot of very important objects.
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Including the gravitational effects of many stellar bodies like planets and stars.
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But also, it turns out that it describes a special type of stellar object called black holes, which we'll come back to in a moment.
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Now, I won't try to describe the exact mathematical form of...
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Because that's not really suitable for this podcast.
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But what we'll do is we'll sort of talk about...
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The key features of the Schwarzschild metric and what it tells us about space-time in in this spherically symmetric.
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Remember, I said that there are four...
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Components to this metric, well, four non-zero components.
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there are 16 in the in the 4x4 matrix, but...
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And the four components each describe essentially how the shape of space and time.
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Is affected by each of the four.
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Each of the four coordinates, one of time and then three of space.
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So two of the spatial ones aren't really, very interesting because of the spherical symmetry.
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The two coordinates of most interest.
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Are the time coordinate, and the radial coordinate.
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So the radius, the radial coordinate, describes how far away we are from the central.
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Being spherically symmetric, what it fundamentally describes, as I said just before.
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Is the effect of a central, spherically symmetrical mass on the space.
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And time around it.
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So this is applicable to planets and the space around them, or stars in the space around them, even galaxies.
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I mean, neither are stars or planets, but...
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To the effect of these central, massive bodies on the space around them.
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Now something interesting if you look at the form of the equation.
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And that both of them involved apparent singularities.
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In this case, it's sort of fairly easy to see because the equation looks like 1 minus some number over r.
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Where R is the radius, the radial coordinate, how far away we are from the central mass.
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So when you have an equation like this, one minus...
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Some number over R.
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If you think about what this means, because R is on the denominator, as R gets bigger, that second term gets smaller.
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realization
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And because there's a minus sign in front of it, it's one minus, then this term that gets smaller, this whole term goes to one.
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As our goes to infinity.
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Now, what this means is that as we move further and further away from the central mass, the metric becomes more and more like the metric just describing flat space.
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1 minus some number over r.
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One over R becomes small.
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And so we're subtracting off a smaller and smaller number.
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What we have left with is closer and closer to one.
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Which is just like the metric for a Flat.
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It's just basically ones along that diagonal, you know.
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So, but this kind of makes sense, essentially, as we get very far away from the central mass.
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The effect of its gravity becomes smaller and smaller.
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Space and time become you more and more like just empty flat space.
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With this mathematical structure, if R gets really, really small.
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Instead of getting big, we're now making really, really small.
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When R gets really, really small, 1 over r...
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Actually approaches infinity, it gets really, really big, and it blows up to, theoretically, infinity in the limit of of are approaching zero.
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So this means that the metric is actually not defined.
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confusion
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When r equals zero.
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And that's what we call the singularity.
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Point well, in this case, it's just sort of in our metric, in the description.
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Where the metric isn't defined.
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The way you can think about this is at R equals 0, the metric is not continuous.
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It sort of, pinches together at a point.
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Just as if you were kind of making a a funnel out of Play-Doh, But at the end, you didn't kind of curve it around.
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But you just sort of pinched it off.
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Uh, sort of rolling up one end, You- you can't just sort of roll around the tip.
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Because it's not continuous, like it ends at a sharp.
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You have to sort of flip your hand all the way around and then go up the other side.
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You meant by a discontinuity.
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