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I want to interpret geodesics in a constant gravitational field.
As a simple example I start with flat Minkowski spacetime
[tex]ds^2 = -dT^2 + dX^2 + dY^2 + dZ^2[/tex]
with a geodesic (in terms of coordinate time T)
[tex]X^\mu(T) = (T, X=A, 0, vT)[/tex]
where A is an arbitrary constant and v ≤ c.
Now I transform this geodesic to Rindler coordinates. These coordinates can be understood as the effect of a constant gravitational field due to a constant acceleration g. However there is the problem that they still describe vacuum spacetime, so they do not provide a gravitational field produced by any non-zero mass distribution (but this is a problem I want to address later).
The transformation reads
[tex]t = \frac{1}{g}\text{artanh}\left(\frac{T}{X} \right) [/tex]
[tex]x = \sqrt{X^2 - T^2}[/tex]
[tex]y = Y[/tex]
[tex]z = Z[/tex]
So for the geodesic we find
[tex]x^\mu(T) = \left(\frac{1}{g}\text{artanh}\left(\frac{T}{A} \right), \sqrt{A^2-T^2}, 0, vT\right)[/tex]
still expressed in terms of T.
Now from x(T) we see that the yz-plane crosses the geodesic with constant X(T)=A at T=A; this can be interpreted as a particle falling in a gravitational field from X=A to the moving yz-plane in coordinate time T=A.
But in Rindler coordinates we find
[tex]t(T\to A-) = \frac{1}{g}\,\lim_{T\to A-}\,\text{artanh}\left(\frac{T}{A} \right) \to \infty[/tex]
So measured in Rindler time t the particle will never cross the yz-plane.
The reason why I work with coordinate time t and T instead of proper time is that I want to include light-like trajectories for v=1. The problem is that I do not see any reference frame S' for which I get a reasonable expression for the coordinate time interval Δt' where the particle falls from the initial position at x'(X=A) to the y'z'-plane with coordinate x'=0.
Remark: what I am really interested in is not the Rindler spacetime but a non-vacuum spacetime produced by a mass distribution with a gravitational field in x-direction which is approximately homogeneous for a region [A-h,A+h]. But first I have to understand the Rindler case.
As a simple example I start with flat Minkowski spacetime
[tex]ds^2 = -dT^2 + dX^2 + dY^2 + dZ^2[/tex]
with a geodesic (in terms of coordinate time T)
[tex]X^\mu(T) = (T, X=A, 0, vT)[/tex]
where A is an arbitrary constant and v ≤ c.
Now I transform this geodesic to Rindler coordinates. These coordinates can be understood as the effect of a constant gravitational field due to a constant acceleration g. However there is the problem that they still describe vacuum spacetime, so they do not provide a gravitational field produced by any non-zero mass distribution (but this is a problem I want to address later).
The transformation reads
[tex]t = \frac{1}{g}\text{artanh}\left(\frac{T}{X} \right) [/tex]
[tex]x = \sqrt{X^2 - T^2}[/tex]
[tex]y = Y[/tex]
[tex]z = Z[/tex]
So for the geodesic we find
[tex]x^\mu(T) = \left(\frac{1}{g}\text{artanh}\left(\frac{T}{A} \right), \sqrt{A^2-T^2}, 0, vT\right)[/tex]
still expressed in terms of T.
Now from x(T) we see that the yz-plane crosses the geodesic with constant X(T)=A at T=A; this can be interpreted as a particle falling in a gravitational field from X=A to the moving yz-plane in coordinate time T=A.
But in Rindler coordinates we find
[tex]t(T\to A-) = \frac{1}{g}\,\lim_{T\to A-}\,\text{artanh}\left(\frac{T}{A} \right) \to \infty[/tex]
So measured in Rindler time t the particle will never cross the yz-plane.
The reason why I work with coordinate time t and T instead of proper time is that I want to include light-like trajectories for v=1. The problem is that I do not see any reference frame S' for which I get a reasonable expression for the coordinate time interval Δt' where the particle falls from the initial position at x'(X=A) to the y'z'-plane with coordinate x'=0.
Remark: what I am really interested in is not the Rindler spacetime but a non-vacuum spacetime produced by a mass distribution with a gravitational field in x-direction which is approximately homogeneous for a region [A-h,A+h]. But first I have to understand the Rindler case.
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