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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. | speaker_0 | neutral | en | |
Hello, you're listening to The Science of Everything podcast, episode 145. | speaker_0 | neutral | en | |
Relativity and black Holes. | speaker_0 | interest | en | |
I'm your host, James Fodor. | speaker_0 | neutral | en | |
This episode is a continuation of the discussion of general relativity, which we began in episode... | speaker_0 | neutral | en | |
136, which is the prerequisite for this episode. | speaker_0 | neutral | en | |
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. | speaker_0 | interest | en | |
And how we combine these formalisms together. | speaker_0 | neutral | en | |
To yield Einstein's field equations, which loosely say that, the curvature of space-time. | speaker_0 | neutral | en | |
Is proportional to the energy and matter content of spacetime. | speaker_0 | neutral | en | |
And I explained how Einstein's field equations are a series of 10. | speaker_0 | neutral | en | |
Coupled nonlinear partial differential equations. | speaker_0 | neutral | en | |
Which means that they're very complex and difficult to solve. | speaker_0 | neutral | en | |
For any realistic cases. | speaker_0 | neutral | en | |
However, I did say that there are some closed form, meaning sort of simply mathematically describable. | speaker_0 | neutral | en | |
Solutions known. | speaker_0 | neutral | en | |
To Einstein's field equations, and I'll talk about them in a future episode. | speaker_0 | interest | en | |
Well, now is that future episode, or. | speaker_0 | neutral | en | |
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. | speaker_0 | interest | en | |
And how it's able to describe Well. | speaker_0 | neutral | en | |
Described the existence of black holes. | speaker_0 | interest | en | |
So we'll talk about deriving the Schwarzschild metric, how to interpret the resulting metric. | speaker_0 | interest | en | |
And then we'll see how. | speaker_0 | neutral | en | |
Z resulting metric. | speaker_0 | neutral | en | |
Yields predictions which have been experimentally verified and thereby serving as experimental evidence in favour of general relativity. | speaker_0 | interest | en | |
We'll then talk in more detail about Schwarzschild black holes. | speaker_0 | neutral | en | |
Some of the phenomena there, like the event horizon, singularity and so forth. | speaker_0 | interest | en | |
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. | speaker_0 | interest | en | |
This will be a pretty dense episode, so hope you're ready. | speaker_0 | neutral | en | |
And let's then jump into it. | speaker_0 | neutral | en | |
But bear in mind, though, I will be assuming... | speaker_0 | neutral | en | |
That you've listened to. | speaker_0 | neutral | en | |
On general relativity, because that introduces some of the key ideas that I'll... | speaker_0 | interest | en | |
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. | speaker_0 | interest | en | |
What we're going to do is try to find a solution to these equations. | speaker_0 | neutral | en | |
There are a number of closed form solutions known. | speaker_0 | neutral | en | |
We're going to focus on one of them today. | speaker_0 | neutral | en | |
And essentially, what this amounts to is solving for the equations to find the metric. | speaker_0 | neutral | en | |
Mathematical description of the overall shape of space and time. | speaker_0 | interest | en | |
A metric that satisfies the equations. | speaker_0 | neutral | en | |
So the term on the left of the equations is the Einstein tensor. | speaker_0 | neutral | en | |
Denoted as a capital, G. | speaker_0 | neutral | en | |
And it more or less describes the curvature of the metric. | speaker_0 | neutral | en | |
Which represents the structure of space and time. | speaker_0 | neutral | en | |
On the right hand side is the stress energy tensile, which describes the energy content of of space and time. | speaker_0 | neutral | en | |
There's sort of two ways to solve. | speaker_0 | neutral | en | |
These couple of equations. | speaker_0 | neutral | en | |
One is to postulate a stress energy tensor, so to stipulate what the energy content of space is, and then solve for the metric. | speaker_0 | neutral | en | |
Given далее energy content. | speaker_0 | neutral | en | |
You can start with the metric. | speaker_0 | neutral | en | |
So specify what, though. | speaker_0 | neutral | en | |
And then solve for the stress energy tensor that will give you that metric. | speaker_0 | neutral | en | |
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. | speaker_0 | neutral | en | |
As well as making a few other assumptions. | speaker_0 | neutral | en | |
And then we're going to see what metric that gives us, what metric satisfies. | speaker_0 | interest | en | |
The equations when we stipulate what the energy content of the universe is. | speaker_0 | neutral | en | |
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. | speaker_0 | neutral | en | |
Is going to be zero. | speaker_0 | neutral | en | |
So we're looking for a vacuum solution, it's often called, an empty space solution of the equation. | speaker_0 | neutral | en | |
So obviously that simplifies things dramatically. | speaker_0 | neutral | en | |
Because the right hand side of the equation is just zero. | speaker_0 | neutral | en | |
And it the equation, simplified down to the the rishi tends, is equal to zero. | speaker_0 | neutral | en | |
Remember, basically describes the curvature. | speaker_0 | neutral | en | |
Of um space of time in that region. | speaker_0 | neutral | en | |
So the next step, now that we've already simplified things quite a lot, is to make further assumptions to help simplify things. | speaker_0 | neutral | en | |
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. | speaker_0 | neutral | en | |
They're constant over time. | speaker_0 | neutral | en | |
So this is just representing the type of solution that we're looking for, which is a static solution. | speaker_0 | neutral | en | |
Again, this is often done for simplicity. | speaker_0 | neutral | en | |
Now it turns out, when you make this assumption, this dramatically simplifies things further. | speaker_0 | neutral | en | |
Because it means that any terms that interact with the time coordinate, Remember, there's four coordinates. | speaker_0 | neutral | en | |
Of, of space and time. | speaker_0 | neutral | en | |
There's one time coordinate and three spatial coordinates. | speaker_0 | neutral | en | |
So any of them that interact with the time coordinate have to go to zero. | speaker_0 | neutral | en | |
There will be changes over time. | speaker_0 | neutral | en | |
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. | speaker_0 | neutral | en | |
And they're just the diagonal components. | speaker_0 | neutral | en | |
So those along the diagonal. | speaker_0 | neutral | en | |
A 4x4 matrix is only the ones along the diagonal. | speaker_0 | neutral | en | |
That will be non-zero, everything else goes to zero. | speaker_0 | neutral | en | |
When we make this assumption of a static field. | speaker_0 | neutral | en | |
Things are now very simple because instead of having these 10... | speaker_0 | neutral | en | |
Coupled partial, different, nonlinear, partial, differential equations. | speaker_0 | neutral | en | |
Now we've got, we've reduced it down to four much simpler equations. | speaker_0 | neutral | en | |
All of which are just equal to zero. | speaker_0 | neutral | en | |
Four equations for each, essentially one for each of the coordinates, a one-time equation. | speaker_0 | neutral | en | |
And one for each of the three spatial coordinates. | speaker_0 | neutral | en | |
Now, to make even more simplifications, we introduce a third assumption, which is that we're going to look for a spherically symmetrical solution. | speaker_0 | neutral | en | |
So we're kind of interested in solutions that look the same, when you rotate them. | speaker_0 | interest | en | |
That simplifies things a lot further because now we only have to worry essentially about two Coordinates. | speaker_0 | neutral | en | |
The radius, which is the distance from the center. | speaker_0 | neutral | en | |
And then one angle coordinate. | speaker_0 | neutral | en | |
Even with all of these simplifications, the equations that we have to solve are still somewhat complicated. | speaker_0 | confusion | en | |
But the algebra is at least now solvable, and obviously I can't go through all the details here. | speaker_0 | neutral | en | |
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. | speaker_0 | neutral | en | |
And then just substitute in for the actual form of the Rishitensa. | speaker_0 | neutral | en | |
Remember, the Rishitensa is defined in terms of mathematical objects called Christoffel symbols. | speaker_0 | neutral | en | |
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? | speaker_0 | interest | en | |
And starting at the North Pole and walking down towards the equator. | speaker_0 | neutral | en | |
Their spear will change direction as they walk along, even if they, it doesn't. | speaker_0 | neutral | en |
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