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Relativity and black Holes. | 3.449 | |
This episode is a continuation of the discussion of general relativity, which we began in episode... | 4.49 | |
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. | 4.507 | |
And how we combine these formalisms together. | 4.487 | |
Which means that they're very complex and difficult to solve. | 4.633 | |
However, I did say that there are some closed form, meaning sort of simply mathematically describable. | 4.494 | |
And then we'll see how. | 3.75 | |
Z resulting metric. | 3.721 | |
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. | 4.634 | |
And let's then jump into it. | 3.998 | |
On general relativity, because that introduces some of the key ideas that I'll... | 4.482 | |
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. | 4.749 | |
What we're going to do is try to find a solution to these equations. | 4.604 | |
There are a number of closed form solutions known. | 4.749 | |
We're going to focus on one of them today. | 4.836 | |
And essentially, what this amounts to is solving for the equations to find the metric. | 4.164 | |
Mathematical description of the overall shape of space and time. | 4.297 | |
A metric that satisfies the equations. | 4.259 | |
So the term on the left of the equations is the Einstein tensor. | 4.486 | |
And it more or less describes the curvature of the metric. | 4.45 | |
On the right hand side is the stress energy tensile, which describes the energy content of of space and time. | 4.38 | |
There's sort of two ways to solve. | 4.366 | |
These couple of equations. | 3.509 | |
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. | 4.22 | |
And then we're going to see what metric that gives us, what metric satisfies. | 4.427 | |
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. | 4.423 | |
Remember, basically describes the curvature. | 4.902 | |
So the next step, now that we've already simplified things quite a lot, is to make further assumptions to help simplify things. | 4.625 | |
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. | 4.002 | |
So any of them that interact with the time coordinate have to go to zero. | 4.602 | |
When we make this assumption of a static field. | 4.339 | |
Now, to make even more simplifications, we introduce a third assumption, which is that we're going to look for a spherically symmetrical solution. | 4.504 | |
And then one angle coordinate. | 3.781 | |
Even with all of these simplifications, the equations that we have to solve are still somewhat complicated. | 4.645 | |
And then just substitute in for the actual form of the Rishitensa. | 4.188 | |
Their spear will change direction as they walk along, even if they, it doesn't. | 4.405 | |
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... | 4.251 | |
That helps us to describe the change in direction of a path over a curved geometry. | 4.204 | |
And so the Rishi tensor is defined in terms of these Christoffel symbols. | 4.817 | |
The metric is the 4x4 array. | 4.412 | |
So we're just going to substitute in the correct forms for each of these components. | 4.21 | |
Additional assumptions of the static field. | 4.261 | |
Do the TDC algebra, rearrange and combine some things together. | 4.816 | |
And this metric is called the Schwarzschild Metric after the... | 4.329 | |
Scientist who derived it originally. | 4.532 | |
But there's a theorem which states that the Schwarzschild metric is the only spherically symmetric vacuum solution of Einstein's field equations. | 4.389 | |
So there's only one solution to Einstein's field equations. | 4.614 | |
That is spherically symmetric. | 4.547 | |
So that's quite an important result, and it turns out that it's not that difficult to derive. | 4.377 | |
Over the decades, there's been a lot of study into this metric and what it tells us. | 4.148 | |
And it turns out that it actually describes a lot of very important objects. | 4.225 | |
Because that's not really suitable for this podcast. | 4.568 | |
But what we'll do is we'll sort of talk about... | 3.97 | |
And the four components each describe essentially how the shape of space and time. | 4.279 | |
Each of the four coordinates, one of time and then three of space. | 4.89 | |
So two of the spatial ones aren't really, very interesting because of the spherical symmetry. | 4.16 | |
Are the time coordinate, and the radial coordinate. | 4.098 | |
I mean, neither are stars or planets, but... | 4.695 | |
To the effect of these central, massive bodies on the space around them. | 4.356 | |
Now something interesting if you look at the form of the equation. | 3.841 | |
In this case, it's sort of fairly easy to see because the equation looks like 1 minus some number over r. | 4.165 | |
Where R is the radius, the radial coordinate, how far away we are from the central mass. | 4.787 | |
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. | 4.022 | |
As our goes to infinity. | 4.633 | |
1 minus some number over r. | 4.712 | |
It's just basically ones along that diagonal, you know. | 3.727 | |
So, but this kind of makes sense, essentially, as we get very far away from the central mass. | 4.114 | |
Space and time become you more and more like just empty flat space. | 4.036 | |
So this means that the metric is actually not defined. | 4.191 | |
And that's what we call the singularity. | 4.3 | |
The way you can think about this is at R equals 0, the metric is not continuous. | 4.278 | |
It sort of, pinches together at a point. | 4.023 | |
Uh, sort of rolling up one end, You- you can't just sort of roll around the tip. | 3.842 | |
You meant by a discontinuity. | 4.12 | |
Continuous sort of curve motion that you can proceeding from one side to the other. | 4.407 | |
Indicate is that there's a breakdown of our theory, the theory is not... | 4.265 | |
Describing accurately what's happening at that very small spatial scale. | 4.665 | |
Now, this isn't really surprising for our approach. | 4.207 | |
General relativity is a description of relatively large distance and time scales, not very small ones. | 4.554 | |
Appropriate to describe whatever's happening at that very small scale. | 4.327 | |
In other words, we know there's a singularity as R approaches 0. | 4.135 | |
It turns out that the Schwarzschild radius, it's quite small for most massive objects. | 4.364 | |
So, most objects in the universe are much larger than their Schwarzschild radius. | 4.688 | |
So for most real physical objects, the Schwarzschild radius doesn't really matter. | 4.271 | |
Because, remember, the Schwarzschild metric only applies in the vacuum, so it's a vacuum solution. | 4.356 | |
Because of the gravitational effects of Earth, but it won't actually describe the space and time. | 4.601 | |
Because obviously that's that's not a vacuum, right? | 3.974 | |
So the the fact that there's sort of weird stuff happening. | 3.911 | |
The Schwarzschild radius sort of never comes into it. | 4.363 | |
You only see the Schwarzschild radius manifested when the vacuum solution is relevant at the Schwarzschild radius. | 4.327 | |
Isn't relevant because the Schwarzschild metric isn't relevant. | 4.672 | |
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... | 4.484 | |
Spherically symmetric stellar object. | 4.69 | |
Only in the case of black holes, do you see... | 4.7 | |
So we'll talk about those a little bit later. | 4.823 | |
It turns out, however, that the apparent singularity that happens at the Schwarzschild radius. | 4.424 | |
It's just because of a poor choice of coordinate system. | 4.779 | |
But there's many coordinate systems you can use to describe the same metric. | 4.525 | |
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. | 4.462 | |
In practice, probably the radial coordinate, plus the latitude and longitude is going to be easier. | 4.136 |
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