What Does The Ricci Tensor Mean? | Tensor Intuition

An intuitive introduction to the Ricci tensor, Riemann curvature tensor, Christoffel symbols, and their roles in general relativity.

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Key Takeaways

The Ricci tensor is derived by contracting the Riemann curvature tensor and is crucial in describing spacetime curvature.
Einstein's field equations relate the geometry of spacetime to the distribution of matter and energy via tensors.
The Schwarzschild metric is a key example of a metric tensor used to describe spherical gravitational fields.
Tensor operations like raising indices and contraction are essential tools in understanding curvature tensors.
The Einstein summation convention simplifies tensor notation and calculations.

What the video covers

Introduction to the Riemann curvature tensor and Christoffel symbols as foundational concepts in differential geometry and general relativity.
Explanation of the Ricci tensor as a contraction of the Riemann tensor and its significance as a rank two tensor.
Discussion of the Einstein field equations, highlighting the interplay between geometry (left side) and matter-energy (right side).
Overview of the stress-energy tensor and its role in describing matter and energy in spacetime.
Description of the Schwarzschild metric in spherical coordinates as an example metric tensor in general relativity.
Explanation of tensor contraction, raising and lowering indices, and the Einstein summation convention for simplifying tensor expressions.
Clarification that the Riemann tensor for the Schwarzschild metric is zero, implying the need for other metrics to observe curvature.
Use of matrix representations to provide intuitive understanding of rank two tensors despite their limitations.
Brief mention of the cosmological constant and its relation to dark energy within the Einstein tensor.
Encouragement to explore linked resources for deeper study beyond the basic overview provided.

Chapters

What is the Ricci tensor and why is it important?

The Ricci tensor is a rank two tensor obtained by contracting the Riemann curvature tensor. It plays a key role in describing how spacetime is curved by matter and energy in Einstein's field equations.

How does the Einstein field equation relate geometry to matter?

The Einstein field equation equates the geometry of spacetime, represented by tensors like the Einstein tensor, on the left side, to the distribution of matter and energy, represented by the stress-energy tensor, on the right side.

Why is the Schwarzschild metric significant in general relativity?

The Schwarzschild metric is a solution to Einstein's field equations describing the spacetime geometry around a spherical, non-rotating mass like a black hole, often used as a foundational example in general relativity.

Full Transcript — Download SRT & Markdown

Speaker A

In this video and the next video, I'm going to give just a brief introduction to the Riemann curvature tensor and the Christoffel symbols. Then, the video after that, I will talk a little bit about the stress-energy tensor, both of which are important in the Einstein field equations of general relativity.

Speaker A

energy tensor both of which are important in the einstein field equations of general relativity so this r here is the reachy tensor which is a rank two tensor this r here is uh the the reachy scalar which i will also talk

Speaker A

So this R here is the Riemann tensor, which is a rank two tensor. This R here is the Riemann scalar, which I will also talk about in this video, and the Christoffel symbols that are actually found from this g here, which is the metric tensor. Same with this g here. This uppercase lambda is actually the cosmological constant, and that has to do with dark energy.

Speaker A

that has to do with dark energy and so this part right here is oftentimes called the einstein tensor and you will often see that as a g so as a capital g mu nu and so its components will be

Speaker A

And so this part right here is oftentimes called the Einstein tensor, and you will often see that as a G, so as a capital G mu nu. And so its components will be the components of the Riemann tensor minus one-half times the curvature scalar there times the metric tensor.

Speaker A

each of these things as usual there are links in the description where you can uh really start digging into these things quite a bit more uh but for this i am just going to give kind of a a really brief basic overview so that

Speaker A

And I'm not going to get too much into the derivation of all this. I mostly just, once again, want to give an intuition of each of these things. As usual, there are links in the description where you can really start digging into these things quite a bit more.

Speaker A

often uh thought about is on the left side of the equal sign here we have the geometry so the differential geometry then on this side on the right side we have uh the the part that talks about the mass and

Speaker A

But for this, I am just going to give kind of a really brief basic overview so that at least if you see this equation, this Einstein field equation, you kind of know what each thing is talking about.

Speaker A

little bit of an idea so this is called the schwarzschild metric it's in spherical coordinates so this is a metric tensor uh and we see that it has this minus this minus 1 minus 2 g m over r c squared and this is the

Speaker A

So this T over here, this second rank tensor, is the stress-energy tensor. And so the way that this equation is often thought about is on the left side of the equal sign here, we have the geometry, so the differential geometry.

Speaker A

a space that looks sort of like this uh and so we see we're flat and then we sort of uh sort of in a symmetrical sense with it and start falling in down towards over here there'd be the

Speaker A

Then on this side, on the right side, we have the part that talks about the mass and the energy. And so the left side is oftentimes said to tell matter how to move, while matter tells spacetime how to be shaped or how to bend or, you know, flex or so forth.

Speaker A

to start looking at that reachy tensor but this is just uh to get sort of an idea of what a metric uh would look like in general relativity all right and so the first thing we have is this riemann curvature tensor

Speaker A

And so down here, just to give you a little bit of an idea, this is called the Schwarzschild metric. It's in spherical coordinates, so this is a metric tensor. And we see that it has this minus, this minus 1 minus 2GM over r c squared.

Speaker A

uh what we would actually want to do is this which is taking this rank four tensor which has one upstairs index and then three downstairs indices uh and then contract it so we we keep the upstairs one i and we change

Speaker A

And this is the same thing, but it's inverse. Then we have this r squared and r squared sine squared theta, which should look familiar from when we're using spherical coordinates to actually look at a sphere.

Speaker A

want so if we look here that has two downstairs indices and so to contract that like i said we are going to sum over i for every value of the h and every value of k so if we have h equals zero and we run

Speaker A

And so it'll actually give a space that looks sort of like this. And so we see we're flat, and then we sort of, in a symmetrical sense with it, start falling in down towards over here. There'd be the singularity of a black hole.

Speaker A

and then again for h equals 3 and i'm not going to sort of talk through all this the the thing to really notice though once again is why we like that einstein summation convention uh and so that is why uh

Speaker A

And actually, with this Schwarzschild metric in spherical coordinates, the Riemann tensor, I believe, is actually zero. So you'd need a different metric, a metric describing something else than this, to start looking at that Riemann tensor.

Speaker A

then these that i have here on the right side of the equal sign i have now put into a matrix that looks like this and this matrix works because if we try to do the the riemann curvature tensor which is

Speaker A

But this is just to get sort of an idea of what a metric would look like in general relativity. All right, and so the first thing we have is this Riemann curvature tensor, which is used more in things like differential geometry, and that's actually a rank four tensor.

Speaker A

more rigorous will say that that will try to often steer away from this matrix representation of of tensors that but i like the matrix representation i think it makes it a little more intuitive uh what's going on but uh yeah since this is only a rank

Speaker A

And so, well, actually here I just have a rank two tensor that I'm contracting down to, well, a rank zero tensor. But what we would actually want to do is this, which is taking this rank four tensor, which has one upstairs index and then three downstairs indices,

Speaker A

the diagonal of this and essentially what we're doing and i kind of gave it away up here uh when we're contracting a rank two tensor is we're taking the trace sort of of the rank 3 tensor or you know

Speaker A

and then contract it. So we keep the upstairs one i, and we change the downstairs one to i, and then we sum over the i. And then that will actually get rid of the two i's, and we will end up with something like R h k.

Speaker A

we're just summing over the t i and the t i so it's just t one one t two two t three three and uh yeah this one i only did a three dimensional rank two tensor but you see

Speaker A

And so that looks more like what we want. So if we look here, that has two downstairs indices. And so to contract that, like I said, we are going to sum over i for every value of h and every value of k.

Speaker A

well and then so if we want to get that riemann scaler well the first thing we have to do is we actually have to raise one of these indices because we can only sum over in upstairs any downstairs and so

Speaker A

So if we have h equals zero and we run through k equals zero, one, two, and three, we end up with this. And then we have h equals one, and we want to run through k equals 0, 1, 2, and 3 again. And then we do it again for h equals 2,

Speaker A

matrix multiplication here and so i just kind of uh put here what matrix multiplication is so to get this uh this element here we actually have to multiply a1 times b1 a2 times b2 and a3 times b3 then to get this one it'd have

Speaker A

and then again for h equals 3. And I'm not going to sort of talk through all this. The thing to really notice, though, once again, is why we like that Einstein summation convention.

Speaker A

uh and then if we just go through and do all the elements uh we end up with something that looks like this but anyway i went through that in more detail in a in an earlier video of how

Speaker A

And so that is why you will see it mostly looking just kind of like this, even without this summation here, that implicit summation when we have two indices that are the same. And so we just run through all of those.

Speaker A

the trace here so just these uh we're just adding these uh these blue r's here and so that's how we actually get this this reachy scalar here so i'm editing this in because i realized that after i recorded i forgot to say what

Speaker A

Then these that I have here on the right side of the equal sign, I have now put into a matrix that looks like this. And this matrix works because if we try to do the Riemann curvature tensor, which is this four-index thing, we'd need a four-dimensional matrix,

Speaker A

try and draw it a little bit better so in a circle in 2d space we have the area of the circle is just everything between these lines here but if we have curved space then we have a circle uh and so the

Speaker A

and that would be quite difficult to try to draw, which is one of those reasons that people who are more rigorous will say that they will try to often steer away from this matrix representation of tensors.

Speaker A

little bit from the side and so it would actually have sort of a a dome on it and so and so the area of that circle is going to be larger if if it's bent because now we have to account for

Speaker A

But I like the matrix representation. I think it makes it a little more intuitive what's going on. But yeah, since this is only a rank two tensor, we can actually draw it in this matrix form.

Speaker A

that's what the the uh scalar curvature is actually telling you geometrically speaking but anyway this uh like i said was edited in after i finished recording the video so now back to our regularly scheduled program all right and then so in the reachy tensor so this

Speaker A

And you can see I put in blue all the ones that are on the diagonal, so the 0 0, the 1 1, the 2 2, and the 3 3, which all go into the diagonal of this.

Speaker A

here going all the way up to here all came from this website which is uh it's not a video it's it's uh something you have to read but it's it's a i found it a really great resource for

Speaker A

And essentially what we're doing, and I kind of gave it away up here, when we're contracting a rank two tensor is we're taking the trace sort of of the rank three tensor, or kind of a trace, because you can see there would be all different kinds of traces we could take by summing over that one instead of that one and so forth.

Speaker A

so up here we just have it with these r's but each element is actually uh is actually this long thing here with all these uh uppercase gammas and some of these partial derivatives so those gammas are the christophel

Speaker A

Where the trace thing works better with a rank two since we're just summing over the t i and the t i, so it's just t one one, t two two, t three three. And yeah, this one I only did a three-dimensional rank two tensor, but you see we're just adding all the diagonals on here.

Speaker A

a metric tensor for two dimensions so i j that's just the e one dot e one e two dot e or e 1 rather e 1 e 2 e 2 e 1 and then e 2 dot e 2

Speaker A

And so that is just the trace. So this is kind of like doing the trace of a rank four tensor in order to generate this rank two tensor.

Speaker A

basis vectors because if we're in a space that is not flat then uh then so you will have at this point uh you know some some basis vectors here so if this is like three dimensions uh but space is curved so you'll curve

Speaker A

Well, and then, so if we want to get that Riemann scalar, well, the first thing we have to do is we actually have to raise one of these indices because we can only sum over an upstairs and a downstairs.

Speaker A

uh we've gone from having z here so maybe we'll call this prime z prime x prime y prime there's some change there's an angle change here in our z and so that's uh and of course also in the x and then the

Speaker A

And so we actually do the index raising, which I talked about in a previous video, in order to sort of raise this mu up to a sigma here. And to do that, we just do this matrix multiplication here.

Speaker A

space here and it's telling us if we went if we start at this point here and we move to this point here and then over to this point here we will have uh we will our basis vector will have changed in

Speaker A

And so I just kind of put here what matrix multiplication is. So to get this element here, we actually have to multiply a1 times b1, a2 times b2, and a3 times b3. Then to get this one, it'd have to be a1 times b2, a2 times b5, and a3 times b8, and so on and so forth.

Speaker A

and so the the this change in the basis vector here this is actually the riemann tensor the richie tensor is the contraction of that but the riemann tensor uh so it's showing us you know some change some difference in those in those uh

Speaker A

And so that's what I actually have showing right here. So this is just looking at how we get this first element.

Speaker A

about it uh is that the reachy tensor and so this is actually the uh riemann tensor the reachy tensor is telling us so uh is telling us essentially how much the uh two geodesics here uh change so if we move say from like right here

Speaker A

And then if we just go through and do all the elements, we end up with something that looks like this. But anyway, I went through that in more detail in an earlier video of how to actually raise and lower indices, so I'm not going to go through that too much here.

Speaker A

like a ball here uh how much the volume of that ball has changed uh or if we have geodesics that you know sort of bend inwards like this we have a ball here and is telling us how much the volume of that ball has

Speaker A

But anyway, the first step is to raise the index, and then we want to contract, which once again is just kind of doing the trace here, so just these, we're just adding these blue R's here.

Speaker A

still from this on this profound physics website and i i like this because uh is the general steps for calculating the reachy tensor so specify a metric tensor uh so we calculate those christopher symbols from the metric which then we

Speaker A

And so that's how we actually get this Riemann scalar here. So I'm editing this in because I realized that after I recorded, I forgot to say what the scalar curvature or Riemann scalar means.

Speaker A

summations explicit on here of the origin tensor so i just took this here and made these summations on each of these things uh explicit um and so we have the this uh christopher symbol these christian symbols here or the derivatives of these

Speaker A

And so what it means is essentially if you have some circle in space here, so a circle in 2D space, the area that's a really bad circle. I'll try and draw it a little bit better.

Speaker A

uh and so that is why the einstein summation is uh is preferred and so down here i have the the christophel symbol at the uh so at this first part right here uh sort of written out explicitly uh so outside the einstein summation

Speaker A

So in a circle in 2D space, we have the area of the circle is just everything between these lines here. But if we have curved space, then we have a circle, and so the space will actually sort of stick out. So...

Speaker A

riemann curvature tensor uh and then the uh that reaches scalar is the contraction of the re or the reachy uh tensor which is a rank two tensor and then the other part is just this idea that the reachy tensor is telling us

Speaker A

how volumes between geodesics which are straight lines in space i mean you see these in the cycle that's not a straight line well it's a straight line but through bent space so it's a space itself that is bent uh not the

Speaker A

straight line so you may have heard of geodesics if we're uh if we have our planet earth here uh and we want to you know go from one place to another you don't just do a straight line you have to do one of

Speaker A

those sort of great circle things over it and that's because that is a straight line on sort of the the bent surface of the earth and so these even though these lines look like they're bent it's actually the

Speaker A

space that is bent and the lines the geodesics are sort of the straight lines through that bent space and so by looking at how a sphere or a circle uh sort of changes in volume between two geodesics tells you

Speaker A

something about how much the space is bent uh but anyway i hope you found this video helpful and uh in the next video like i said i will talk a little bit more in depth about the christophel symbols and sort of how to

Speaker A

uh intuitively interpret those uh and uh yeah anyway i will see you in the next video you

Topics:Ricci tensorRiemann curvature tensorChristoffel symbolsEinstein field equationsstress-energy tensorgeneral relativitySchwarzschild metrictensor contractioncosmological constantdifferential geometry

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