Re: What is the shape of the Rindler horizon?

From: David McAnally (D.McAnally_at_i'm_a_gnu.uq.net.au)
Date: 01/09/05 

Date: 9 Jan 2005 02:50:12 GMT

D.McAnally@i'm_a_gnu.uq.net.au (David McAnally) writes:

<snip>

>The trajectory of the accelerating observer is given by

> t = 1/g sinh(gs),

> x = 1/g cosh(gs) - 1/g,

> y = 0,

> z = 0,

>where g is the constant acceleration, and is in the positive x direction,
>the observer is stationary at the origin at time t = 0, s is the proper
>time that has passed for the observer since t = 0, and c has been set
>equal to 1.

>The light that reaches the observer at proper time s starts at t = 0 from
>a sphere with centre (1/g cosh(gs) - 1/g, 0, 0) and radius 1/g sinh(gs).
>As s approaches infinity, the radius approaches infinity, and the sphere
>approaches the plane x = - 1/g. This plane is the Rindler horizon.

This plane is the Rindler horizon at t = 0.

The light that reaches the observer at proper time s starts at t = T from
a sphere with centre (1/g cosh(gs) - 1/g, 0, 0) and radius
1/g sinh(gs) - T. As s approaches infinity, the radius approaches
infinity, and the sphere approaches the plane x = T - 1/g. This plane is
the Rindler horizon at t = T.

In other words, the Rindler horizon at t = T is the plane x = c T - c^2/g.

I would like to make an extra comment in the light of earlier discussion
in the thread "The clash of titans", regarding the contrast between
Minkowskian geometry and Lobachewskian geometry. The appropriate geometry
for spacetime is Minkowskian geometry for 3+1 dimensions. One
3-dimensional generalization of Lobachewskian geometry is representable by
the set

        H = {(x,y,z) in R^3 : x > - 1/g},

with metric given by

        ds^2 = (dx^2 + dy^2 + dz^2)/(x + 1/g)^2.

As noted above, the light that reaches the observer at proper time s
starts at t = 0 from a sphere with centre (1/g cosh(gs) - 1/g, 0, 0) and
radius 1/g sinh(gs). These spheres are all within the set H, and the
spheres in the Lobachewskian geometry which are represented by these
spheres are concentric spheres (i.e. concentric in the Lobachewskian
space) with the common centre being the point represented by (0,0,0), and
with the sphere in Lobachewskian space, which is represented by the sphere
from where the light that reaches the observer at proper time s starts at
t = 0, having radius gs. This is not to say that there is any
significance whatsoever in the coincidence. I just thought that the
coincidence was interesting. That is, I thought that it was interesting
that the sphere from where the light that reaches the observer at proper
time s starts at t = 0 represents, in the Lobachewskian geometry, a sphere
with centre (0,0,0) and radius gs.

David

-----

Relevant Pages

  • Re: Fit sphere to planar circles
    ... My approach so far has been to use hough circle detection in the area of = ... sphere regions of the sphere, and I am using some logic = ... I have a similar problem and I abandoned the hough transform. ... normalization though) at the radius of interest. ...
    (comp.soft-sys.matlab)
  • Re: What is the shape of the Rindler horizon?
    ... >As s approaches infinity, the radius approaches infinity, and the sphere ... This is the most general form of the trajectory, up to rotations in space, ...
    (sci.physics.relativity)
  • Re: Question about analysis of Schwarzschild solution on Fo. of Phys. 1988, 18, 6
    ... "On the gravitational field of a sphere of incompressible fluid ... for the mass point. ... The Schwarzschild radius. ...
    (sci.physics.relativity)
  • Re: average distance between points inside or on a sphere
    ... between two points in a ball of radius R. ... correspond to a circular slice through the sphere with radius r2; ... the intersection with the surface of the sphere of a line ... element divided by the total surface area of the sphere, ...
    (sci.math)
  • Re: Gravitational potential inside a sphere
    ... >My book gives for the gravitational potential inside a sphere of radius, ... >with no hint of how it was derived. ... I took it from r= 0, but if taken from infinity you have the added ...
    (sci.physics)