General Relativity in Geometric Algebra 4: Einstein’s Equations

Transcript

All right, welcome to general relativity in geometric algebra episode 4 Einstein's equations. So to discuss Einstein's equations, we must first derive the curvature tensors. The first tensor is the remon tensor and it arises naturally when considering curvature. You might remember from other series that curvature is typically computed using the commutator of a derivative or gradient and this is exactly how we get the remon tensor. Specifically, is my Okay, my pen's working.

Specifically, we consider the following. We consider M, which is just an arbitrary multi vector that is in the space-time algebra. So now we say that M is just some multi vector. We consider the covariant gradient wedged with the covariant gradient all times m or rather than times this is acting on this the coariant derivatives here are acting on the multiv vector m. So we can we can expand this and this is just going to be g mu into we can expand it into the coordinate basis g mu d mu all acting on m.

And then we can factor these out into the wedge product because these are scalers. Recall these are scalers. And this right here is just the these are the actual vectors. So we can factor out the vectors into a wedge product and we can factor the partial or these coariant derivatives into a um a commutator because the when the vector B when the basis of the object commute or is anti-commuting with another basis object then also so are the coefficients but recall recall just real quick I'm going to put a little thing here. There's a little definition is that the commutator of two things A and B is 1/2 A minus B A.

In our in our convention, we're using this factor of 1/2 because it's more natural in geometric algebra. I don't know if you can hear that but the VA auto vacuum is emptying itself really loud. Okay. So we can when we factor it into two things because of this factor of 1/2 we want to cancel it. So we have a factor of two.

We have G mu wedge G new and then times that we have the commutator of these two objects and then all of this is acting on the multi vector M. So how are we going to uh continue with this? We actually have to first expand this. So we must now expand two times the commutator of the covariant derivatives acting on some multi vector m. So if we just copy this because I don't want to write it twice. This is just going to be d mu du minus d new d mu acting on m this in the center.

And then we can plug this in d mu. And then just for clarity's sake, I'm going to emphasize that they're acting each to on each other successively. So this acts first on m then this acts on the entire thing. And likewise here. And so these objects even though even though this has a scalar component it can commute it can anti-commute or it can have a non-commutative part acting on m because it's a derivative.

So it's an operator not just some not just some blade. Okay. So now we can plug in the definition of the covariant derivative for both of these. So we're going to have d mu acting on partial partial new m plus then omega new m minus and I'm just going to copy this whole thing and change the coefficients. So now it's new mu mu.

And then we're going to do this one more time. We're going to plug in the definition here and here. So we get a fairly lengthy expression. As you'll see, we get partial mu partial mu m plus partial mu of the entire um what do you call it? the entire uh commutator and then plus the commutator omega mu partial mu m then plus the commutator of a commutator omega mu omega new m and then we're going to have minus partial Mu partial mu m minus partial mu and then omega mu m minus omega mu partial mu m then minus and then we have one last commutator but I'm going to move everything over here so that it fits minus the commutator of omega mu with the commutator omega mu m. As you can see, this is a fairly lengthy expression, but we're going to simplify things in order to get a more workable a more workable version of this because doing all these terms seems pretty complicated.

Well, the first thing to notice is actually this and this cancel out because partial derivatives commute with each other. And so the derivative by definition of partial derivatives, these are the same object. And so you have it minus itself, it's zero. So we can say but partial mu partial mu minus or yeah m minus partial mu partial mu m= z by definition. So now we must separately expand the following.

We must separately expand [Music] um oh shoot this minus this and then this or this term this term minus this term So now we're going to separately expand these uh terms in the series or not in the series but in the in this equation right here. This one's actually the easiest to expand. Well, it's not like they're actually both relatively easy. They just uh this one has more steps in it, but this one is you get it in just uh two lines essentially. So or two or three lines.

So first we have partial mu of omega mu m then plus omega mu partial mu then minus partial new omega mu m minus omega new partial mu m and then I'm just going to give a little bit of motivation for uh maybe seeing what the answer is being able to see what the answer is without actually doing all this math because if you know if you recognize the patterns you'll actually be able to just go directly to the answer from this and that's because when you take the derivative of this entire thing, you do the product rule where if you have say, you know, the derivative of a then that's equal to the derivative acting on a and then plus the derivative acting on b. And so you have those two objects. And so you can imagine that you're going to get partial mu, you're going to get an object that's just has partial mu acting on m. And if that's the case, then it's going to it would have a positive sign and this has a negative sign. And that's exactly this.

So you're going to just have that this term will cancel with this ter um one of the terms from here will cancel with this and then one of the terms from here will cancel from with this. So you should be able to just predict that from from looking at it now but I'll expand it out just so you can see it explicitly. So expanding this out, the first term is d mu omega mu m because recall that these are scalar. So plus omega mu partial mu m. You can already see here's the term that will cancel with this.

And so what we're going to actually do is copy and paste it right here so that it's really obvious that they're cancelling. And then we're going to just have uh minus partial or going to expand this term minus partial mu omega mu m minus omega mu partial mu m. And then this is just the this this is just the complement to this other term. So we're just going to copy and paste that in. Show that those directly cancel.

And so you get the only the only things left over are this partial mu omega mu m plus the partial or sorry minus the partial mu omega mu m. However, the commutator product is uh distributive and so we can just combine this into a single commutator and that's going to be d mu omega mu minus d mu omega mu. So I'm commuting with m. Let me put this there. Okay.

So, this is the first um thing we needed to expand. And now we're going to do the second one. So, second, we're going to expand this term. And this has more steps, but it's pretty straightforward actually. So that we're going to use we're going to directly go or insert this definition here.

So we're going to get a factor of 1 /2 1 over two. So one 1 over4. Why do we have 1 over4? It's because we have two commutators. So I'm going to apply that definition immediately to take us to this form of 1 over4. And it's going to now be I'm going to have this time this entire commutator and this and this entire commutator.

So omega mu the entire commutator of omega new mus omega mu. Then it's going to be minus omega mu m plus or minus m omega mu omega mu. So that's the first the first term and we can just copy and paste this. Nope. opy and put a minus plus and then swap all of the indices.

Um, so it goes new mu mu mu mu new new. Okay, so that's the the first uh expan expansion. So you can already see that it's pretty it's a simple expansion, but there are lots of terms. So we get 1 over4 and we're just going to distribute through. So we get first omega mu omega new m - omega mu m omega new and then we're going to get minus omega new m omega mu plus m omega new omega mu and then minus omega new omega mu U m plus omega new m omega mu then uh plus omega mu m omega new and then min - m omega mu omega new And now we're going to have a bunch of terms cancel.

Namely, all of these sandwich, all these sandwich terms will cancel. So this mu m new cancels with this mu. This new mu cancels. This then cancels with this term. So now we're just going to be left with these um uh the the quadratic terms, not the sandwiching terms.

though technically a sandwich is a quadratic but don't mind me. So now we just have omega mu omega mu m plus or I'm going to just to emphasize I'm now going to bring the term with m on the right side over here. So now we have omega mu omega mu m and then plus or rather I'll do minus m omega mu omega new plus m omega new omega mu. So now I'm going to apply the definition of a commutator again. I'm going to do the reverse thing.

So again using this equation. So now we just have a factor of 1 /2 and we're going to have commutator of omega mu and omega mu * m and then minus m * omega mu omega new And you'll notice that this is just the definition of another commutator. So we can apply this again and we get the commutator of the commutator omega mu omega new m. So we can this is the second thing we wanted to show. the second thing.

So now we can move on to our original task which was expanding just the commutator of the gradients or not gradients the derivatives. So therefore because of one and two we have 2 * the commutator of the covariant derivatives acting on some multi vector m gives us and We're just going to copy. So, oh, that's not the just copy this. And then this. Uh, I don't like that positioning like this.

Okay, that's better. It's It's still going a little down, but whatever. We can now simplify this because recall both of these are commutators and now they're acting both on m so it's distributive. So we can combine this into a single commutator that is I can do this a little cleaner. uh d mu omega mu d mu omega mu plus the commutator of omega mu omega new.

All of this acting on m via a commutator. And this is the definition of the remon tensor mu new with m. So this is the remon tensor specifically it is a b vector. So this is the reman by vector and you can see you can recall from the past video where these omegas were telling you or here I'll just go up to the definition of the omega. These omegas were telling you how the coordinate frames were changing based on the or how how they were changing.

So these kind of contain the the curvature in them. um they don't give you the curvature directly but they contain the information of the curvature. To find the curvature you find out how going in different directions. Now if you have two of these you have two of these connection by vectors and you see how they change with respect to each other both in the derivatives and then with how they commute then this gives you the total curvature and so that is why this is then the remon tensor which is the curvature tensor or the it's a b vector in this case and don't let these mu news get you confused this is not the traditional form for the remon tensor. I will give you that in a second.

But these munus do not correspond to the traditional munus in the remon tensor. So let's talk a little bit about this and it's going to be this. So the remon tensor is defined as R with two with a B vector argument and this is just more simply denoted with R mu. So notice that these indices aren't the uh like tensor indices. They're indices that correspond to the B vector argument going in and what the coordinate directions are.

Okay, so that's the definition of it. But then we can also just plug in this. And how would this look in the traditional form? Well, to do that, we need to express it in a bi vector basis. And that would then just be this. So we get r mu then alpha beta g alpha wedge g beta.

And so now we've expressed this is a scalar and this is the bi vector. So because this is scalar this is now the actual traditional remon tensor. And these alpha beta indices are the indices that you would normally see in traditional um general relativity. So just for emphasis I'm going to I'm going to make a better arrow than that. This is the traditional [Music] remon tensor.

Just going to go a little closer. And just for sake of consistency, we can also express this in the tetrad basis. It It's funny. uh the source that I used to learn all of this was like you can see how in the tetrad basis that the alpha beta indices anti-commute but they do in this one too because this is also a bi vector so I just found that a little funny it's like yeah these do anti-commute because this is a bveector um it's just one of the while it was a good source it has a lot of typos in it and there were some just and and some errors where that you can see they might be new to geometric algebra Anyway, there's also not just the remon tensor, there's going to be something called the Richie tensor. So, there's also the Richie tensor.

Richie tensor, which pops out. pops out. When considering the curvature of a or rather along a coordinate basis G mu. So we take this we take this to consider the curvature acting on G mu. And this then gives us the G mu wedge G m new of R mu new G mu or rather G new.

So I'm just going to put this new here. So we get this and by definition this is the inner product between a by vector and a vector. So this returns a vector. So we get g mu which g new and this is r mu new dotted with g new. Oops.

Comment. There we go. And we can simplify this such that we have g mu new g mu wedge gu and then this is times r mu. So this r mu is the richy tensor which is also called the richy vector. Now so just like this can be called the remon bi vector.

This is the richy vector. So this implies or actually let me put it in the center. This implies that r mu is defined as the remon bi vector with one of its indices dotted out. then uh something or hang on I I didn't do the sake of I didn't do the consistent thing. It's R defined as G with a single argument or R with a single argument.

So now or with a a vector argument not a B vector argument. So even though R kind of defines it looks like it might be the same thing as the remon tensor. Notice that the that the remon tensor has a B vector argument. So this is a B vector. Whereas this has a vector argument.

So this is a vector. And there's also the Richie scaler, the Richie Scaler. And that is typically written with a curly R. And it's defined as the inner product between G mu and the richie vector. We can also express this in terms of the remon by vector.

Okay, that was interesting. I didn't know that when I receive a call, it cancels my recording. So um yeah, I'm going to have to edit those two bits together. But yes, we can also express this in terms of the remon bi vector. So we then have to expand this out.

So we get r mu newu dotted with g new. And then we can use the fact that this is anti-commutative to then bring this over like so. And then we can then pull this out to have an outer product between these two. And then the it'll be the inner product of two bi vectors together. And because of the minus sign, this G new is actually going to go out onto the left hand side.

And so we end up with G new wedge. Actually we end up with G new wedge G m new dotted with R mu new. And so you can express the Richie scaler as the inner product of the Richie vector and a coordinate basis or the you can do that with the W or the the B vector the coordinate by vector basis uh dotted with the Richie or sorry the remon bctor. Okay. And so now we have all of the curvature tensors that we need to discuss Einstein's equations.

Let's summarize this though. So therefore we have we have the following. So we have the remon we have the remon by vector that is defined as r mu m new which is just the derivatives and the commutators of the connection by vectors. Then we have the Richie vector which is just R mu and that's just going to be R mu new dotted with G new. can do a better new.

And then we have lastly the Richie scaler and that is just going to be cursive R which is R mu or sorry it's going to be G mu dotted with the Richie vector or we can have G mu wedge g mu dotted with the remon bi vector. So now we can talk about what Einstein's equations are after we then define the Einstein tensor. The Einstein tensor and really it's a vector. So the Einstein vector is as follows. It is g mu equals just the richy vector r mu minus 12 then the coordinate g mu of all times the richy scalar.

So this is the Einstein tensor and you can also represent this in a coordinate free basis and that is going to be g of a. So if a is a vector this is a vector in spacetime then this is going to be equal to the re the richy vector r of a min - 1/2 this vector and then the richy scalar which will also depend on this a. So the Einstein tensor or the Einstein vector is defined as this and now we get to the Einstein equations. So if t mu which is a vector. So if t mu and just to emphasize that it's a vector we have one three and if this is the stress energy tensor and this is just a vector.

So the stress energy tensor is just a vector. Then Einstein's equations R and I say R but really it's one equation and it's just this it's G mu equals kappa and this This is just some uh constant coefficient. I'll explain it in a second. Is equal to the stress energy tensor plus the coordinate basis time lambda. And we'll talk about what the what these are in a second.

And if you want to express this in the coordinate free form, then it would be g of a is going to be equal to kappa * t of a and then plus a lambda. Again, we'll talk about what this lambda is in a second. square kappa which is equal to 8 pi g where this g is even though it has the same notation as the einstein tensor this g is actually Newton's constant of gravitation and all of that over c to 4th and this is approximately 2.077 077 * 10 -43rd and this is in units of 1 / newtons and this is the Einstein coefficient the Einstein or constant rather the Einstein constant and lambda here is the cosmological constant And so these are the Einstein equations for uh general relativity in geometric algebra. It's just telling you that the curvature G. So the Einstein tensor is just the curvature.

It's telling you that the curvature is equal to some proportion of the stress energy of the system plus some constant. And originally when Einstein created this he he added this term so that he could artificially define the universe as static and not expanding not not contracting. But then when it was discovered uh that there was expansion it turned out that this needed to be used even though Einstein at some point went back and just got rid of this. And so at some point when or when Einstein originally proposed his equations, there was this factor and then he got rid of it and said it was the biggest mistake of his life and then it turned out that it was necessary and he was right to have it in the first place just for a different reason. So this is what accounts for the fact that the universe is accelerating and its expansion.

Uh but yeah, these are the it's just literally saying that the the vector that tells you where all of the energy is going is going to manifest itself as the curvature and that's how Einstein uh Einstein led to this revolution in in gravitation where it's now viewed as the curvature of space which is rather elegant working in the space-time algebra getting this as a result. And let's now just give an example of what the stress energy tensor could be. So an example of tu t mu is t mu = 1 / 2 f g mu f reverse where F is a B vector in the space-time algebra and it is the electromagnetic field. And so you can imagine that it's actually relatively easy to solve these equations in geometric algebra. And that's I don't know about easy, but that's what we'll cover in the next video.