Levi-Civita connection and pseudoRiemannian metric

[martinbn]

Just out of curiosity, what is the point of this thread?

 

[PeterDonis]

In the text it is literally stated "we need to calculate the null geodesic from G1 to us".

Which, taken in context, means "find an expression for $\int ds = 0$ along the null geodesic, and then split it up into pieces that have different physical interpretations". What the text does *not* say is anything along the lines of "these separate pieces, even though they have obvious physical interpretations as integrals along timelike or spacelike geodesics, are actually integrals of some function other than $ds = 0$ along the null geodesic the light follows".

Calculations based on the integration of the null geodesic equations

I'm not sure what you mean by "integration of the null geodesic equations". Finding an expression for $\int ds = 0$ is not the same as integrating the geodesic equation; the $\int ds = 0$ expression can be written down purely from the FRW line element. The geodesic equation itself doesn't appear anywhere in the integrand.

Hairsplitting whether thy are about "how far" or "how long" or whether each integral by itself is the integral of a null geodesic is a purely terminologic discussion that is of not much utility in practice.

If you mean you don't need to go into all these details in order to get a numerical answer, of course you're right. But the same can be said of hairsplitting about whether what we are doing can be interpreted as calculating a "distance" along the null geodesic that's different from the physical $ds = 0$ distance (i.e., whether we can interpret what we're doing as calculating $D_T$ instead of $D_M$), as opposed to splitting up the $\int ds = 0$ expression into pieces with obvious physical interpretations as integrals along timelike or spacelike geodesics. None of these interpretations matter if all you're concerned with is getting a numerical answer for $D_p$. They only matter if you have particular preferences for how to justify interpreting $D_p$ as a "distance".

 

[TrickyDicky]

Which, taken in context, means "find an expression for $\int ds = 0$ along the null geodesic, and then split it up into pieces that have different physical interpretations". What the text does *not* say is anything along the lines of "these separate pieces, even though they have obvious physical interpretations as integrals along timelike or spacelike geodesics, are actually integrals of some function other than $ds = 0$ along the null geodesic the light follows".

I am saying(like you I think) that they are integrals of $ds = 0$ along the null geodesic the light follows.

I'm not sure what you mean by "integration of the null geodesic equations". Finding an expression for $\int ds = 0$ is not the same as integrating the geodesic equation; the $\int ds = 0$ expression can be written down purely from the FRW line element. The geodesic equation itself doesn't appear anywhere in the integrand.

Right, the sentence is read:integration of the "null geodesic" equations (ds=0), not integration of the null "geodesic equations".

If you mean you don't need to go into all these details in order to get a numerical answer, of course you're right. But the same can be said of hairsplitting about whether what we are doing can be interpreted as calculating a "distance" along the null geodesic that's different from the physical $ds = 0$ distance (i.e., whether we can interpret what we're doing as calculating $D_T$ instead of $D_M$), as opposed to splitting up the $\int ds = 0$ expression into pieces with obvious physical interpretations as integrals along timelike or spacelike geodesics. None of these interpretations matter if all you're concerned with is getting a numerical answer for $D_p$. They only matter if you have particular preferences for how to justify interpreting $D_p$ as a "distance".

Fair enough.
Since there is some curiosity about the point of this thread and just in case you are interested in what goes on beyond the purely computational side of it, the chapter "On the structure of space-time" page 178(in google books preview) in the book "Space, time and geometry" is what inspired my OP.

 

[martinbn]

Do you mean the book by P. Suppes? I must say that I have absolutely no idea how that page of the book led you to this question!

 

[TrickyDicky]

Do you mean the book by P. Suppes? I must say that I have absolutely no idea how that page of the book led you to this question!

Yes, that book. But that is just the first page in the essay, he mentions metric spaces and how this definition applies to relativity and classical space-times in pge 180.