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Continuous sort of curve motion that you can proceeding from one side to the other.
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You have to sort of sharply stop and then kind of rotate around.
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And that's thought to be non-physical, that sort of singularity there, because space-time is thought to be continuous.
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So singularities like this sort of these sharp discontinuities in space-time, are not thought to be.
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well, many people think that they're not real.
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Indicate is that there's a breakdown of our theory, the theory is not...
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confusion
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Describing accurately what's happening at that very small spatial scale.
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interest
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Now, this isn't really surprising for our approach.
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Approaches zero, because it when I get small enough, what this means is that our distance scale is getting small enough.
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And when it becomes...
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The effects of quantum mechanics will become relevant.
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And we don't have a theory of quantum gravity.
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General relativity is a description of relatively large distance and time scales, not very small ones.
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And so it's not really surprising that the theory breaks down at very small scales, because we kind of know that it's...
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Not going to be suitable at that scale.
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disappointment
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So these are two interesting things about the Schwarzschild metric is that when you go very far away from the central mass, the metric just looks like out of flat space.
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And when you go right to the center, or as you approach the center, there is a singularity.
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So the metric becomes undefined, telling us that our theory won't be...
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confusion
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Appropriate to describe whatever's happening at that very small scale.
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There's another interesting feature of the Schwarzschild metric, which is that is that when you solve the equations, you find that there's this sort of special length scale where there's apparently another singularity.
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In other words, we know there's a singularity as R approaches 0.
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But there's another place where it looks like there's a singularity in the equations, where R approaches this special length scale.
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This special length scale is being given a name.
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It's called the Schwarzschild radius.
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It turns out that the Schwarzschild radius, it's quite small for most massive objects.
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So, most objects in the universe are much larger than their Schwarzschild radius.
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The Schwarzschild radius of any object is determined entirely by its mass.
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So, for example, the Schwarzschild radius of Earth is about one centimeter.
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The Schwarzschild radius of the Sun is about 3 kilometers.
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So for most real physical objects, the Schwarzschild radius doesn't really matter.
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Because, remember, the Schwarzschild metric only applies in the vacuum, so it's a vacuum solution.
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So the Schwarzschild Metric will describe the space and time outside of Earth.
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Because of the gravitational effects of Earth, but it won't actually describe the space and time.
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Inside the Earth itself, or inside the sun itself, right?
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Because obviously that's that's not a vacuum, right?
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There's, there's matter there.
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So the the fact that there's sort of weird stuff happening.
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confusion
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At the Schwarzschild radius, and, and.
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Below that, down to the singularity.
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Is irrelevant for most stellar objects, because most stellar objects are much larger than the Schwarzschild radius.
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The Schwarzschild radius sort of never comes into it.
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Because, again, the Schwarzschild radius will only exist in space.
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When the object is smaller than the Schwarzschild radius and the Schwarzschild radius is exposed.
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And therefore, the vacuum actually exists, where the Schwarzschild radius is.
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You only see the Schwarzschild radius manifested when the vacuum solution is relevant at the Schwarzschild radius.
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Because the Schwarzschild metric is only relevant for a vacuum solution if you've got matter there than the Schwarzschild radius.
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Isn't relevant because the Schwarzschild metric isn't relevant.
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Now, there are special types of objects for which the Schweitzer radius is relevant, because the Schwarzschild radius is, so to speak, exposed to space, And therefore, it does describe...
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In that area, and these are called black holes.
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We'll come to these in a moment.
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But the Shrachio metric doesn't just describe the space and time around black holes, it describes the space and time around any spherical So...
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Spherically symmetric stellar object.
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Only in the case of black holes, do you see...
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interest
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The Shwashu radius and other sort of bizarre phenomena.
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confusion
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So we'll talk about those a little bit later.
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It turns out, however, that the apparent singularity that happens at the Schwarzschild radius.
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It's just because of a poor choice of coordinate system.
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There's different coordinate systems that we can use to describe the same metric.
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So the Schwarzschild metric, there's only one Schwarzschild metric.
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But there's many coordinate systems you can use to describe the same metric.
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Just as I can describe.
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A location on Earth's surface, by using some kind of XY coordinate system, or I can use distance from the center of the earth.
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And angle, Well, I guess I need two angles, right?
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But I could use the distance from the center of the Earth, and then I could use two angles to describe the position.
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On the surface of the Earth, essentially, that would be latitude and longitude, but if I wanted to, I could use a simple cartesian XYZ.
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Both of those could be used to describe position on the surface of the Earth.
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In practice, probably the radial coordinate, plus the latitude and longitude is going to be easier.
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But either of them describes the same set of positions.
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And so it's the same thing with the Schwarzschild metric.
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There's only one Schwarzschild metric, but there are many ways to represent it using different coordinates.
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That the apparent singularity at the Schwarzschild radius isn't real.
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realization
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There's nothing, there's no actual discontinuity of spacetime there.
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It just appears to be.
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Discontinuity because of poor choice of coordinates.
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disappointment
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In different coordinate systems, you don't see any singularity there.
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There's still something important happening at the Schwarzschild radius, as we'll describe later, but it's not actually a singularity.
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So there's an important difference between the parent singularity at the Schwarzschild radius.
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interest
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Which is due to our coordinate system.
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If we choose a different coordinate system to represent the same Schwarzschild metric, then that singularity goes away.
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And it's just a regular...
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Point in space-time, like any other one.
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It does still have important properties as we'll talk about.
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In a moment, but it's not.
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confusion
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Space-time is still continuous there.
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Whereas the singularity at R equals 0 is a real geometric singularity.
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It exists in any coordinate.
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System for describing the Schwarzschild metric so you can't transform it away by using different coordinates.
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interest
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And it seems like...
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neutral
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That, according to general relativity, space-time is pinched there and is not continuous.
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And so that's an indication that our theory breaks down there.
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Now, there's some other interesting aspects of interpreting what the Schwarzschild metric is telling us about space and time.
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interest
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I'm apart from the singularity and the Schwarzschild radius.
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Easiest way to understand...
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interest
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What's happening here is that the mass at the centre.
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Generating the distortion that's described by the Schwarzschild metric, that central mass distorts space and time.
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And it distorts them in a particular way, specifically...
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It elongates essentially radial distances, or distances to and from.
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And it also elongates temporal distances.
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Are specifically described in terms of proper time.
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So that time duration is measured by an observer.
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