- #1

space-time

- 218

- 4

ds

^{2}= (cdt)

^{2}- R

^{2}(t)[dr

^{2}/(1- kr

^{2}) + r

^{2}(dθ

^{2}+ sin

^{2}(θ)dΦ

^{2})]

Here is the metric tensor:

g

_{00}= 1

g

_{11}= - R

^{2}(t) / (1- kr

^{2})

g

_{22}= - R

^{2}(t)r

^{2}

g

_{33}= - R

^{2}(t)r

^{2}sin

^{2}(θ)

Every other element is 0.

Now from this tensor, I know that I can calculate the proper times and proper distances between events in this space time using the formula ds

^{2}=g

_{μν}dx

^{μ}dx

^{ν}. I can also gain info on whether two events in this metric are space-like separated, time-like separated, or light-like separated.

That is the info that I can gain about the space time using the metric tensor. (Feel free to let me know if I am missing anything).

Now let us skip to the Ricci tensor:

R

_{00}= - 3R''(t) / R(t) where R''(t) is the 2nd derivative of R(t) with respect to t.

R

_{11}= [2(R'(t))

^{2}+ R(t)R''(t) + k] / (1- kr

^{2})

R

_{22}= 2r

^{2}(R'(t))

^{2}+ r

^{2}R(t)R''(t) + 2kr

^{2}

(Note: The term r and the function R(t) are not the same. I am just saying this to avoid confusion between the r and R)

R

_{33}= 2r

^{2}(R'(t))

^{2}sin

^{2}(θ) + r

^{2}R(t)R''(t)sin

^{2}(θ) + 2kr

^{2}sin

^{2}(θ)

Every other element is 0.

Now, what information do I actually get from the Ricci tensor (in general, but you can apply this question to the Ricci tensor that I just derived as an example)? Here is what I mean when I ask this: You know how I was able to derive proper times and proper distances using the metric tensor? Well what information about events in a metric (or the structure of the space time itself) can I derive from the Ricci tensor? Please be specific and technical. I know it tells you something about the curvature of space time, but what is that something? As things stand right now, I only know that a non-zero Ricci tensor implies that the metric has curvature (and even then it can't even be said that a 0 Ricci tensor implies no curvature because the Ricci tensor is just a contraction of the Riemann tensor which may have some non-zero elements that simply aren't covered in the Ricci tensor). I know that with the Riemann tensor, you can find the change in a vector's orientation if you parallel transport it around a manifold, but I don't think you can do that with the Ricci tensor.

Now let us head to the curvature scalar:

The curvature scalar = [ - 6R''(t)R(t) - 6(R'(t))

^{2}- 5k] / R

^{2}(t)

The same question that I had for the Ricci tensor applies to the curvature scalar. What info do I get from it? All I know about it for right now, is that a non-zero curvature scalar means that the metric has curvature, and that a 0 curvature scalar means that the metric is flat space.

Now for the Einstein tensor:

G

_{00}= [6(R'(t))

^{2}+ 5k] / (2R

^{2}(t))

G

_{11}= [ - 4R''(t)R(t) - 2(R'(t))

^{2}- 3k] / (2 - 2kr

^{2})

G

_{22}= [ - 4r

^{2}R''(t)R(t) - 2r

^{2}(R'(t))

^{2}- kr

^{2}] / 2

G

_{33}= [ - 4r

^{2}R''(t)R(t)sin

^{2}(θ) - 2r

^{2}(R'(t))

^{2}sin

^{2}(θ) - kr

^{2}sin

^{2}(θ)] / 2

Every other element was 0.

As you may expect at this point, my question remains the same. What information can I extrapolate with this tensor? Is there any formula containing the Einstein tensor that I can use to derive some info about events and curvature in space-time like there is with the metric tensor?

Now here is a big one that I have been trying to grasp:

I haven't yet derived the stress energy momentum tensor for this particular metric because I am still trying to find out whether or not this metric has a non-zero cosmological constant, and if so what that constant is (if anybody can tell me this then that would be nice).

However, with other stress energy momentum tensors that I have derived, I have noticed that even when I derive the tensor itself, I can never seem to get any info from it.

I know the meanings of the various elements of the tensor. I know that T

_{00}

is energy density, the other elements in the temporal row and the temporal column are momentum density, and the rest of the elements are momentum flux.

However, it is the energy density of what? The momentum flux of what? The matter/energy in the metric? Is it the energy density of the matter in the metric at any given event in the space time, or is it the energy density of matter required for the creation of such a region of space time? How exactly do I get useful information from the stress energy momentum tensor? I've asked in threads before if quantities such as the T

_{00}in a stress energy momentum tensor dictate the energy density that it takes to create such a space time. Every time I have asked if this was the correct interpretation, I have been told that this was not how to interpret it.

To make matters worse, whenever I look at papers on that particular metric, they don't even seem to focus on these tensors that much, as they more so seem to focus on vectors and co-vectors that aren't even in the Einstein field equations and seem to come out of nowhere.

For example, when I did the Godel metric, the paper talked more about the rotation vector ω

^{μ}and the fact that T

_{μν}= ρu

_{μ}u

_{ν}where ρ is the density of the matter in the metric and u

_{μ}is the co-vector that corresponds to the 4-velocity of the matter in the metric. It didn't even tell me how ρ was derived or where it came from. How would I know what the density of the matter in the metric is when the sites that I go to in order to find the metrics that I study usually only give me the line element? They don't even give me the cosmological constant. Not to mention the fact that according to the wiki, the formula T

_{μν}= ρu

_{μ}u

_{ν}only works with dust solutions to the EFE's. Other media goes on to mention killing vectors and other differential geometry topics that don't appear in the Einstein field equations.

It almost feels as though the tensors in the EFE's are useless because I never seem to get any info out of them except for what I get from the metric tensor.

This is why I ask, what information about space- time curvature and events do the tensors in the Einstein field equations actually give you? Please be specific. You can use the metric shown in this thread as an example to explain your points if you want.

Thank you