Second order expansion of metric in free-fall

In summary, the metric in a freely-falling frame can be expressed by the formula ds2 = -c2dt2(1 + R0i0jxixj) - 2cdtdxi(\frac{2}{3} R0jikxjxk) + (dxidxj(δij - \frac{1}{3} Rikjlxkxl) to second order. The derivation of this result involves expanding the metric in a Taylor's series and choosing coordinates such that the Christoffel symbols vanish. Under certain assumptions, this simplifies to Cμστν = (1/4)Rμσντ.
  • #1
InsertName
27
0
Hello,

I have read that, in a freely-falling frame, the metric/ interval will be of the form:

ds2 = -c2dt2(1 + R0i0jxixj) - 2cdtdxi([itex]\frac{2}{3}[/itex] R0jikxjxk) + (dxidxjij - [itex]\frac{1}{3}[/itex] Rikjlxkxl)

to second order.

Does anyone know where I could find a derivation of this result?
 
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  • #2
InsertName, That's an interesting formula! It's derivation is almost immediate, except I have doubts about the absence of the time coordinate, and the factors of 1/3.

Expand the metric in a Taylor's series:
gμν = Aμν + Bμνσxσ + Cμσντxσxτ + ...
It's always possible to choose coordinates such that Aμν = ημν and Bμνσ = 0, and in these coordinates the Christoffel symbols vanish. Then the formula for the Riemann tensor reduces to
Rμσντ = ½(gμτ,σν + gσν,μτ - gμν,στ - gστ,μν) = Cμστν + Cτνμσ - Cμσντ - Cσμτν.
If you assume C to have the same symmetry as the Riemann tensor, then this is 4Cμστν, showing that Cμστν = (1/4)Rμσντ
 

1. What is the second order expansion of metric in free-fall?

The second order expansion of metric in free-fall is a mathematical concept used in the study of general relativity. It involves analyzing the behavior of a metric tensor, which describes the curvature of spacetime, in a region where gravity is the dominant force. The second order expansion takes into account the effects of both the gravitational field and the acceleration of the observer in free-fall.

2. How is the second order expansion of metric in free-fall different from the first order expansion?

The first order expansion only takes into account the effects of the gravitational field on the metric tensor, while the second order expansion also includes the effects of the observer's acceleration. This makes the second order expansion more accurate and applicable in situations where the gravitational field is strong, such as near massive objects like black holes.

3. What are some applications of the second order expansion of metric in free-fall?

The second order expansion is used in the study of gravitational waves, which are ripples in the fabric of spacetime caused by accelerating masses. It is also used in the analysis of the motion of objects in strong gravitational fields, such as the orbits of planets around stars or the trajectories of spacecraft near massive objects.

4. How is the second order expansion of metric in free-fall calculated?

The calculation of the second order expansion involves using mathematical equations and techniques from differential geometry and tensor calculus. It can be a complex process, but modern computer programs and simulations have made it easier to perform these calculations and analyze the results.

5. Is the second order expansion of metric in free-fall always necessary?

No, the first order expansion is often sufficient for many practical applications. However, in extreme situations where the gravitational field is very strong, the second order expansion may be necessary to accurately describe the behavior of objects in free-fall. It also provides a more complete understanding of the effects of gravity on spacetime.

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