I What's the underlying frame of the Einstein's Field Equation?

[Pyter]

The chart you are looking at is a global chart, not a local one.

I don't understand what you mean by "local" or "global" chart. To me a chart is a set of functions mapping the N+1 coordinates of the ambient (embedding) space to the N local coordinates of the N-dimensional manifold.
Here's another example of that type of charts: http://berkeleyscience.com/gr.htm
Why should you use curvilinear coordinates for T** and G** when you have the linear, local coordinates at your disposal?

 

[PeterDonis]

To me a chart is a set of functions mapping the N+1 coordinates of the ambient (embedding) space to the N local coordinates of the N-dimensional manifold.

That's not what a "chart" is in any GR textbook I've read, certainly not in any of the references you've been given, or in any of the Insights articles you've been referred to, or in the examples you yourself have been posting. In the example you've given, Schwarzschild coordinates in a spherically symmetric spacetime, the spacetime itself is a 4-dimensional manifold and you have 4 coordinates, $t$, $r$, $\theta$, $\phi$. Each point in the spacetime is labeled by a 4-tuple of coordinates, with appropriate properties of continuity. There is no "mapping" to any embedding space anywhere.

Since the chart covers the entire spacetime (at least in the example you're discussing--note that this would not be true in the case of a black hole, but you should not even try to understand that case until you understand the simpler case of a spherically symmetric massive body surrounded by vacuum, which is what the example you've been posting is describing), it is global. A "local" chart in GR is something like the Riemann normal coordinates I described earlier; the coordinates are "local" because the expression for the metric in these coordinates is exactly valid only at the single chosen point on which the coordinates are centered, and becomes more and more inaccurate as you get further away from that point. The expression for the metric in Schwarzschild coordinates, however, is valid everywhere.

Why should you use curvilinear coordinates for T** and G** when you have the linear, local coordinates at your disposal?

You do not have "linear, local coordinates" at your disposal, not the way you are using that term. You do not have any embedding of spacetime in any higher dimensional Euclidean space. The examples you have given of embeddings are not of spacetime, they're of 2-surfaces in Euclidean 3-space. Those examples do not generalize to spacetime. They just don't. If you continue to insist on treating spacetime this way, this thread will be closed as that is not the way it is done in the GR literature and we cannot help you if you insist on continuing to try to understand it that way. It won't work.

 

[Pyter]

If you look at the Insights articles or the textbook references that have been given, you will see specific solutions being derived using the same general method.

It would take me weeks, I hoped to have a quicker answer.
So far what I gleaned from this thread is: you can choose whatever type of coordinates you see fit. Of course the resulting g** will vary accordingly, I might add.
The only thing I'm missing is how to link the physical observation of the mass points to the setup of the EFE in the general case.

 

[PeterDonis]

Here's another example of that type of charts: http://berkeleyscience.com/gr.htm

You seem to have a talent for picking bad sources. I see no point in trying to deconstruct what is wrong with this one when we have given you a number of good sources already.

 

[Pyter]

You seem to have a talent for picking bad sources. I see no point in trying to deconstruct what is wrong with this one when we have given you a number of good sources already.

It's based on Richard L. Faber, "Differential Geometry and Relativity Theory", Marcel Dekker, 1983.
I guess also that one goes in the trashcan.

 

[PeterDonis]

It would take me weeks, I hoped to have a quicker answer.

As Euclid is supposed to have said to a king who wanted a quicker answer from him, there is no royal road to learning.

You do realize that those of us experts who are trying to help you here have spent years or decades studying GR, right?

So far what I gleaned from this thread is: you can choose whatever type of coordinates you see fit. Of course the resulting g** will vary accordingly, I might add.

Yes, that is correct.

The only thing I'm missing is how to link the physical observation of the mass points to the setup of the EFE in the general case.

Ok, here's the fully general case. Pick coordinates $x^\mu$, where $\mu$ ranges from $0$ to $3$. The general expression for the metric in these coordinates is:

$$
ds^2 = g_{\mu \nu} x^\mu x^\nu
$$

where $g_{\mu \nu}$ is a symmetric, 2nd-rank tensor with 10 independent components. Each of those 10 independent components is an unknown function of the four coordinates. (I have used the Einstein summation convention in the above, so the expression on the RHS is actually a sum of 10 terms in the general case.)

Now just compute the Einstein tensor of the above and set each of its 10 independent components (which will be a differential equation in the $g_{\mu \nu}$ and their first and second derivatives) equal to the corresponding component of the SET, which will be another function of the four coordinates, which might be known or unknown. (If $g_{\mu \nu}$ appears in the SET, that just becomes part of the unknown function of the four coordinates for that SET component.)

You now have a system of 10 second order differential equations for (possibly, depending on how many SET components you know) up to 20 unknown functions of the four coordinates. Then all you have to do is solve it. Good luck.

 

[PeterDonis]

It's based on Richard L. Faber, "Differential Geometry and Relativity Theory", Marcel Dekker, 1983.
I guess also that one goes in the trashcan.

Are these the only textbooks you have read? They're both by mathematicians. Have you read any textbooks by physicists? Such as, oh, I don't know, the two classic textbooks in the field that I've already referenced (MTW and Wald)? Or the nice introductory lecture notes by Carroll, another physicist?

 

[PeterDonis]

It's based on Richard L. Faber, "Differential Geometry and Relativity Theory", Marcel Dekker, 1983.
I guess also that one goes in the trashcan.

Based on the seriously mistaken understanding these books appear to have given you, yes, they should go in the trash can.

 

[Pyter]

Are these the only textbooks you have read? They're both by mathematicians. Have you read any textbooks by physicists? Such as, oh, I don't know, the two classic textbooks in the field that I've already referenced (MTW and Wald)? Or the nice introductory lecture notes by Carroll, another physicist?

As I've answered in another thread, I also read the Dirac textbook, but it's very criptic even if it covers a lot of subjects that others don't, like the EM field component of the SET.
I have the MTW in my library but it's more than 1000 pages thick, I think I'll go with the Wald.
BTW are you little biased towards mathematicians or is it just an impression of mine :)?

Now just compute the Einstein tensor of the above and set each of its 10 independent components (which will be a differential equation in the and their first and second derivatives) equal to the corresponding component of the SET, which will be another function of the four coordinates, which might be known or unknown. (If appears in the SET, that just becomes part of the unknown function of the four coordinates for that SET component.)

Ok but how the (at least initial) observations of the masses enters the picture, as in the solar system scenario I've brought up?

 

[PeterDonis]

You now have a system of 10 second order differential equations for (possibly, depending on how many SET components you know) up to 20 unknown functions of the four coordinates. Then all you have to do is solve it. Good luck.

Of course no physicist actually does this. That's why asking for "the general case" of solving the EFE is pointless.

What physicists actually do is look for various different ways of reducing the number of unknowns in the above scheme. The most obvious way is to assume various symmetries for the spacetime. For example, spherical symmetry, as I said before, reduces the number of unknown functions in the metric to two; and furthermore, even if we make no other assumptions, we know that those two functions can only be functions of the $r$ and $t$ coordinates, not the angular coordinates (assuming we are using standard Schwarzschild coordinates). And it turns out that in this general case there are only three independent components of the EFE, so we have three independent components of the SET that are possibly unknown functions. That reduces us to three differential equations for five unknown functions, so all we have to do is make some specific assumption for two of the unknown functions and we have a well-defined solution. (What I have just said is a quick summary of the Insights article of mine on the Einstein Field Equation for a spherically symmetric spacetime.)

But of course there are many other possible symmetries one could try; or one could try other ways of reducing the number of unknowns, by making other restricting assumptions. So there is not just one "general case" of solving the EFE. There are lots and lots of different special cases, and the best way to understand how they work is to actually go look at them and work with them, one at a time. The references we have already given are a good start.

 

[PeterDonis]

I have the MTW in my library but it's more than 1000 pages thick, I think I'll go with the Wald.

If you have a more mathematically rigorous bent, Wald is probably a better choice anyway. There is lots of math in MTW, but in general their approach is less rigorous and more geared towards physical intuition.

BTW are you little biased towards mathematicians or is it just an impression of mine :)?

There's nothing wrong with math, but it's not physics. If you want to learn physics, you should learn it from physicists, just as if you want to learn math, you should learn it from mathematicians. Mathematicians are just as horrified at physicists' attempts to teach math as physicists are at mathematicians' attempts to teach physics.

 

[PeterDonis]

how the (at least initial) observations of the masses enters the picture, as in the solar system scenario I've brought up?

Please read my post #70 first before trying to respond to my description of the general case.

 

[PeterDonis]

how the (at least initial) observations of the masses enters the picture, as in the solar system scenario I've brought up?

Assuming you've read post #70, here goes.

To analyze the solar system, we make a number of restrictive assumptions to reduce the number of unknowns. In the simplest case, where we consider the Sun to be the only massive body and the planets to be test objects, and we assume the Sun to be non-rotating and therefore spherically symmetric, we have already reduced the number of unknown functions in the metric to two, and since we are assuming vacuum outside the Sun itself (and we're not trying to solve for the Sun's interior), we have no unknown functions at all in the SET, since all of its components are just zero. And we already know that in this case, there is just one unique solution of the EFE, the Schwarzschild solution, in which the metric components are just functions of $r$, and there is really only one such function that has to be determined by solving the EFE, which turns out to be $1 - 2M / r$ (with $M$ the Sun's mass), since the same function appears in both $g_{tt}$ and $g_{rr}$ (just in the denominator of the latter). (The theorem that proves this is called Birkhoff's theorem, and the proof is discussed in the Insights article on it that I referenced.)

Notice that nowhere in any of this did any "initial conditions" appear. That's because this solution is static, so there is no need for any "initial conditions" for the solution; it's the same at all times. The initial conditions for the motion of the planets is only needed to solve the geodesic equation for the orbits of each planet, not to solve the EFE itself. (I already pointed this out to @cianfa in an earlier post.)

 

[Nugatory]

Intuitively I thought that the very concept of "curvature" implied the embedding on a higher space.

It does not - you are confusing two very different concepts: intrinsic curvature and extrinsic curvature.

The surface of a two-dimensional sphere (such as the idealized surface of the earth) has intrinsic curvature. The interior angles of triangles sum to more than 180 degrees, the Pythagorean theorem does not in general work, there are no parallel "straight lines" (these are great circles) because any two will always intersect at two points, ... All of these curvature effects are observed without ever leaving the two-dimensional surface of the Earth and are calculated and observed without ever involving the third dimension. More formally, the Riemann tensor is non-zero everywhere (and note that this is a coordinate-independent statement). Indeed, the whole point of differential geometry is to provide a mathematical treatment of N-dimensional manifolds without embedding them in a higher-dimensional space.

Extrinsic curvature is what happens when we bend an N-dimensional surface through a higher-dimensional space without distorting its geometry. Everyone's favorite example is the cylinder formed by bending a two-dimensional sheet of paper through the third dimension and joining the edges to form a tube: the surface remains euclidean and the hypothetical two-dimensional inhabitants of the sheet of paper will find none of the locally measurable effects that we find on the surface of a sphere. Formally, the Riemann tensor is zero everywhere - no curvature.

If you only have a 1D space, you wouldn't have "curved" or "straight" lines, you'd only have "lines", because the curvature only appears in a 2D space or higher. Same thing for the 2D, and higher dimensions, surfaces.

It is true that a 1D space cannot have intrinsic curvature (it can have extrinsic curvature - just form a closed circle and as with the two-dimensional tube no local property is affected), that the curvature of a 2D surface is characterized by a single number at every point, and so forth. But this has nothing to do with embedding in a higher dimension; we get these results by looking at the degrees of freedom of the Riemann tensor for the N-dimensional space we're considering. Which brings us to...

After all, the curvature of a surface is the rate of change of its normal vector

This is just plain wrong and may underlie much of your confusion here. The intrinsic curvature of a surface is defined by the Riemann tensor, and the Riemann tensor is defined by the behavior of infinitesimal vectors parallel transported within the manifold; no additional dimension is involved. That "rate of change of the normal vector" applies to extrinsic curvature and is completely irrelevant to general relativity.

 

[Pyter]

No, it isn't. It's a global coordinate on the spacetime. The chart you are looking at is a global chart, not a local one.

We're talking about the same thing. What I call "local coordinates" are what you call "global coordinates, labels attached to the spacetime events". I'm referring to the coordinates, each one $\in E_1$, that a "local" chart covering all the manifold (thus actually a global chart) maps to, according to this definition:

To Waner's credit, he also notes that:

@PeterDonis I think I've got the solar system case. The link between the physical observation and the SET is very simple.
What about a more complex case, where you have multiple comparable masses in the same region? What kind of constraints should the physical observations put on the EFE?

That "rate of change of the normal vector" applies to extrinsic curvature and is completely irrelevant to general relativity

Somebody should tell it to this author:

The Rhijk are the components of the Riemann curvature tensor. Note that the Riemann curvature tensor is defined in terms of the Christoffel symbols, which are functions of the metric coefficients. The Lij and Lij are functions of how the normal vector U changes, i.e., the curvature of the surface. Gauss defined the curvature of a 2-d surface geometrically, and showed that it equals det([L])/det([g]). The Riemann curvature tensor allows the curvature to be computed from values that can be calculated in the surface, the gij. It provides the building blocks for the left side of Einstein's field equations.

 

[Pyter]

The intrinsic curvature of a surface is defined by the Riemann tensor, and the Riemann tensor is defined by the behavior of infinitesimal vectors parallel transported within the manifold; no additional dimension is involved.

Just for the records, I've checked Dirac's General Theory Of Relativity and he also seems to adopt the approach of the immersion in the N+1 dimensional space when explaining the parallel transport:

 

[PeterDonis]

What I call "local coordinates" are what you call "global coordinates, labels attached to the spacetime events".

Ok. That means the reference you give is using terrible terminology, different from the literature on GR written and used by physicsts.

Somebody should tell it to this author:

No, somebody should, as I've already said, stop trying to learn physics from mathematicians. Physicists won't bother trying to tell the author this because none of them are going to read his book anyway. And mathematicians won't bother because they're not physicists and don't see the disconnect in the two bodies of literature.

That said, there is a key point that your reference fails to mention. It has chosen a particular embedding for the examples it gives of manifolds that makes the extrinsic curvature (the "rate of change of the normal vector") numerically equal to the intrinsic curvature (the Riemann tensor). But that does not make the two concepts identical, nor does it mean they will always be numerically equal for any embedding.

I strongly suggest that you take the time to read some references by physicists about GR, such as the ones we have already pointed out.

 

[PeterDonis]

I've checked Dirac's General Theory Of Relativity and he also seems to adopt the approach of the immersion in the N+1 dimensional space when explaining the parallel transport:

Now check the references we gave you. What do they say?

Also note that Dirac does not say that $N$ dimensional curved spacetime is embedded in $N + 1$ dimensional flat space. He says that 4-dimensional spacetime can be treated as if it is embedded in $N$ dimensional flat space, without specifying what $N$ is. If you work it out, you will see that $N$ is not 4+1 in the case of 4-dimensional spacetime; it is greater (I think it works out to be 10 in the general case). Not all curved $N$ dimensional manifolds can be embedded in a flat $N+1$ dimensional space; that happens to be true for 2-dimensional manifolds but it does not generalize to higher dimensions.

 

[Pyter]

Now check the references we gave you. What do they say?

For instance Carroll says:

I was citing Dirac because he's a physicist and he talks about manifolds embedded into higher dimension spaces, that you so much abhor. Just sayin'.
We all agree that the EFE is not expressed in ambient coordinates.

 

[PeterDonis]

We all agree that the EFE is not expressed in ambient coordinates.

Yes, that was going to be the next thing I pointed out: that since your concern in this thread is solving the EFE, you should be looking at how all of the references discussed do that. They all do it in coordinates on the manifold itself, not making use of any embedding.

 

[PeterDonis]

I was citing Dirac because he's a physicist and he talks about manifolds embedded into higher dimension spaces, that you so much abhor.

My point all along is that if you want to solve the EFE, looking at embeddings is the wrong way to do it. Of course the embeddings your references talk about exist mathematically. That doesn't mean they're useful for doing physics.

 

[dextercioby]

As with Feynman and Weinberg's work on GR, Dirac's value of his 75 page brochure comes from treating GR as a classical field theory.

 

[vanhees71]

Besides, I really don't know how I could infer the general case from that special case.

Intuitively I thought that the very concept of "curvature" implied the embedding on a higher space.
If you only have a 1D space, you wouldn't have "curved" or "straight" lines, you'd only have "lines", because the curvature only appears in a 2D space or higher. Same thing for the 2D, and higher dimensions, surfaces.
After all, the curvature of a surface is the rate of change of its normal vector, and that is "outside" the surface, in a higher dimension space.

No! It's crucial to understand that you can define a Riemannian (or for GR a pseudo-Riemannian) manifold without any reference to an embedding in an higher-dimensional Euclidean (or pseudo-Euclidean) affine (flat) space. This has been found already by Gauss when he investigated 2D surfaces embedded in 3D Euclidean affine space: You don't need the embedding space to describe all the intrinsic properties of the surface, including distances and angles as well as curvature.

For a straight-forward introduction to GR, I'd recommend Landau&Lifshitz vol. 2. It contains the utmost minimum of tensor calculus you need to understand GR (Ricci calculus and tensor components wrt. holonomous coordinates) and concentrates on the physics. For the more refined mathematical aspects as well as more modern mathematical tools, you may look at Misner, Thorne, Wheeler, Gravitation.

 

[Pyter]

I've realized there's a big piece info I'm missing to get the whole picture: what coordinates system exactly is used for astronomical distances?

From my sparse notions, it should be a (quasi?)-Cartesian system with the z axis passing through the North Pole and the xy plane intersecting the Ecliptic line.
Of course a local chart in the tangent space can't be used because its Euclidean/Minkowskian approximation is only acceptable in a small region of the spacetime.

Does this frame take the curvature into account somehow?

 

[PeterDonis]

what coordinates system exactly is used for astronomical distances?

The usual ones are FRW coordinates, of which there are several different versions, described here:

https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric#General_metric

The usual ones are not Cartesian because our astronomical observations are of angular positions of objects on the sky, with distances estimated from those and other observations. So spherical coordinates of some kind (with several different possibilities for the radial coordinate) are much more suitable.

 

[PeterDonis]

Does this frame take the curvature into account somehow?

Of course. Any valid coordinate chart on a curved spacetime that covers more than a local patch will have to take curvature into account in its expression for the metric.

 

[Ibix]

The usual ones are FRW coordinates,

That depends on what you mean by "astronomical distances". FRW is for very large distances (as I know you are well aware). We mostly just use good old Euclid for the solar system, probably with polar coordinates centered on something useful, or Sun-centered Schwarzschild if we need GR corrections. Presumably you'd use some modified Schwarzschild coordinates (or you could keep the coordinates and modify the metric) to model the solar system with planets as gravitational sources.

 

[PeterDonis]

That depends on what you mean by "astronomical distances".

I had assumed that @Pyter meant distances on cosmological scales, but perhaps he can clarify. Or perhaps he meant both.

 

[PeterDonis]

We mostly just use good old Euclid for the solar system, probably with polar coordinates centered on something useful, or Sun-centered Schwarzschild if we need GR corrections.

AFAIK the usual solar system coordinates are barycentric, so all objects, including the Sun, move in these coordinates. The GR corrections are small enough in the solar system that you don't need the full Schwarzschild metric to model them; just the first few post-Newtonian orders is enough.

Presumably you'd use some modified Schwarzschild coordinates (or you could keep the coordinates and modify the metric) to model the solar system with planets as gravitational sources.

I believe that gravitational corrections due to the planets are modeled as Newtonian forces from point sources, not by altering either the solar system coordinates or the overall metric. The general framework used for these kinds of multi-body problems for weak fields is the Einstein-Infeld-Hoffman equations:

https://en.wikipedia.org/wiki/Einstein–Infeld–Hoffmann_equations

 

[Pyter]

Of course. Any valid coordinate chart on a curved spacetime that covers more than a local patch will have to take curvature into account in its expression for the metric.

I gather that all these coordinates systems assume that the metric is already known?
For instance in the FRW the curvature is uniform everywhere:

where

ranges over a 3-dimensional space of uniform curvature, that is, elliptical space, Euclidean space, or hyperbolic space.

so for "local" variations of curvature, like in a relative small region with several comparable masses, would it still be accurate?

our astronomical observations are of angular positions of objects on the sky, with distances estimated from those and other observations. So spherical coordinates of some kind (with several different possibilities for the radial coordinate)

Always in the case of many masses in a small region, wouldn't the radial coordinates be a little off because of the gravitational lens effect?

I had assumed that @Pyter meant distances on cosmological scales, but perhaps he can clarify.

I was thinking more of a relatively small region with comparable and meaningful masses, like maybe a neutron stars field, or just two orbiting one around the other.