[math, computation] Raychaudhuri equation in time-like geodesic

This is a English version of my former post which was in Korean.

Today i wanna discuss the derivation of Raychaudhuri equation which you will encounter this topic when you study about Singularity theorem and cosmology.

[Documentary on Amal Kumar Raychaudhuri, the renowned theoretical physicist from Kolkata]

Raychaudhuri is a Indian physicist who study fundamental problems of general relativity. The purpose of this post is to show some computation about his named equation not explanation about his life, so i just attach some documentary on youtube and continue the computation.

Generally we decompose geodesic into three parts, time-like, space-like, and null. (which depends on its square is either -1, +1 or zero on the convention of mostly positive convention)

Since time like and space like are just nothing but a different sign of norm vector, their approach are same(one tangent vector is enough) but for null by introducing another auxiliary null vector the case is slightly different. So in this post we are talking about time-like(same approach can be done for space-like) and in the next post we will talk about null case.

The original approach is work on time-like geodesic and later, Sachs expand this to null version.

As i mention before, first in this post we will talk about time-like version. [space like is exactly same except its space-like vector is 1 (not -1) and its metric decomposition changes to h-uu ] Here i follow the general treatment in the textbook and some review papers.

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When you study fluid mechanics, you often encounter the concept of expansion, shear, rotation in vector flow. This equation classify these effect and show the dynamics of matter in vector flow.

Today i wanna derive this equation

This equation plays a crucial role in proving Penrose, Hawking and Georch's Singularity theorem. And further more imposing strong or weak energy condition we can obtain some information about early universe. i.e. via this equation we can study cosmology.

Convention and notation

Here we use mostly positive metric in d=4 (-,+,+,+) is time-like unit vector field which satisfies (For the space-like vector instead of -1 we plug +1)

Decomposition of vector

Now let us decompose

The reason why i repeat the same thing in many ways is to express the physical quantity B_{ab} into trace-less symmetric part and anti-symmetric part and trace part. We can do the similar treatment in Rimeann geometry, in that theory, we can decompose Riemann tensor into Weyl tensor and Ricci tensor and Ricci scalar.

First we can simply decompose this into symmetric and anti-symmetric. Since anti-symmetric naturally gives trace-less, we have to check whether symmetric part is symmetric. Since we have no guarantee that u_{a;a} to be 0, so we have to exclude this term by adding or subtracting some trace term.

To do that we decompose metric g into 3-dimensional metric and u^a.

Directly, contract with u^a we can see

i.e., h_{ab} is a 3-dimensional metric orthogonal to u^a. with this metric h

Explain in more detail, this sigma term represent the "shear" which is symmetric and trace-less part and orthogonal to u^a. and w represent rotation part which is anti-symmetric and orthogonal to u^a. And lastly theta is the expansion which is trace part and orthogonal to u^a.

Since h is orthogonal to u, we are left with checking sigma is orthogonal to u directly. .

During the computation we used.

Frankly speaking, if we did not impose geodesic condition, then we have to slightly changes the term in sigma and w and this term produce so called acceleration term. For convenience here we will impose geodesic condition.

Curvature

From the definition of Riemann curvature for time-like vector u

Note we see the term of B^2, and from the symmetric argument of contraction of symmetric and anti-symmetric indices we have

Now

Up so far, we did not use any thing except geodesic condition. Now by using equation of motion (Einstein field equation), and imposing our matter is a perfect fluid we can do

plugging these results we have

which we want to prove

This is a first order differential equation. The initial condition is related with the early universe so it is often used to study cosmology.

For Null case the coefficient 1/3 changes to 1/2 but the approach are slightly different.
I'll introduce this in my next post