I have been using this paper to study the properties of the Godel metric:(adsbygoogle = window.adsbygoogle || []).push({});

http://www.math.nyu.edu/~momin/stuff/grpaper.pdf

I have some questions that start from Theorem 3.1 and go from there.

1. On page 5, it says that G_{μν}= R_{μν}- ½ g_{μν}R = 8πGρu_{μ}u_{ν}+ g_{μν}Λ (note: Λ is supposed to be the cosmological constant, also u_{μ}is a covariant 4- velocity vector component and the expression ρu_{μ}u_{ν}equals the stress energy momentum tensor T_{μν}) (There must have been a typo in one part of the paper where it has ρu^{μ}u^{ν}instead of ρu_{μ}u_{ν}. Please correct me if you think they actually meant to use superscripts the first time for whatever reason instead of subscripts.)

This basically boils down to writing the Einstein field equations as:

R_{μν}- ½ g_{μν}R = 8πGT_{μν}+ g_{μν}Λ

instead of the usual

R_{μν}- ½ g_{μν}R + g_{μν}Λ = 8πGT_{μν}

Is there some kind of special circumstance where you are supposed to put the cosmological constant on the right hand side of the equations and a special circumstance for the left hand side? For example, is there some rule such as: If the cosmological constant is negative then you put it on the right, but if the constant is positive then you put it on the left? If there is not some kind of rule like that, then why and how did the writer of the paper know to put it on the right instead of the left?

2. The paper says that T_{μν}= ρu_{μ}u_{ν}. Now, according to my calculations, if you do some algebraic manipulation of this version of the EFEs:

R_{μν}- ½ g_{μν}R = 8πGT_{μν}+ g_{μν}Λ

then you can derive the stress energy momentum tensor as follows:

T_{00}= 1/(8πG)

T_{02}and T_{20}= e^{x1}/(8πG)

T_{22}= e^{2x1}/(8πG)

All other elements are 0.

Note: This uses the c= 1 convention, Λ = -1/(2a^{2}) , ρ = 1/(8πGa^{2})

Now if you set ρu_{μ}u_{ν}equal to T_{μν}then you should be able to derive the covariant 4-velocity vector components:

u_{0}= a

u_{2}= ae^{x1}

The other two components are 0.

Here is an example of how I calculated these components:

T_{00}= ρu_{0}u_{0}=

1/(8πGa^{2}) * u_{0}* u_{0}= 1/(8πG)

For this equation to hold true, then u_{0}* u_{0}must equal a^{2}, so then u_{0}= a

That is how I calculated my covariant 4-velocity vector. Now I successfully derived the covariant 4-velocity vector. However, the conclusion of this section of the paper said that the particles in the Godel space time actually have velocities that correspond to the contravariant version of this 4 velocity vector:

u^{μ}= <1/a , 0 ,0 ,0>

Now it is a simple step to simply raise an index for a vector, so I understand how this was derived.

However, why did the velocity vector get changed from the covariant to the contravariant version in the first place? In other words, why do the particles in a Godel space-time have velocity vector u^{μ}instead of u_{μ}? After all, it is u_{μ}(and not u^{μ}) that plays a part in calculating T_{μν}. Would it be wrong to say that particles in a Godel space-time travel with velocity vector u_{μ}?

3. Finally, what exactly is a? How do I determine its value?

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# I have some questions about the work in this paper

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