Schwarzschild radial light rays

However, the outward directed ray from outside the r=2m line will always remain outside the singularity, "escaping" the black hole. This is due to the fact that the singularity is a point of infinite curvature, so an outward directed ray will continue moving away from it.
  • #1
kikitard
6
0

Homework Statement



Consider Schwarzschild spacetime.
A) Show that the equation for ingoing/outgoing radial light rays is dt/dr = +-r/(r-2m) in t,r coordinates and dt*/dr = -1, dt*/dr=(r+2m)/(r-2m) in t*,r coordinates
B) Sketch the local light cones in t,r and t*,r coordinates
C) Explain which coordinates give a truer physical description of the local light cones.
D) Explain the motion of a radial light ray emitted near r=2m.


Homework Equations


schwarzschild metric 0=g[itex]_{\mu\upsilon}[/itex](x(t))[itex]\dot{x}[/itex][itex]^{\mu}[/itex](t)[itex]\dot{x}[/itex][itex]^{\upsilon}[/itex](t)= -[itex]\frac{r-2m}{r}[/itex] +[itex]\frac{r}{r-2m}[/itex][itex]\dot{r}[/itex][itex]^{2}[/itex]+r[itex]^{2}[/itex]sin[itex]^{2}[/itex][itex]\theta\phi[/itex][itex]^{2}[/itex]+r[itex]^{2}[/itex][itex]\dot{\theta}[/itex][itex]^{2}[/itex]

t[itex]_{*}[/itex]=t+2mln([itex]\frac{r}{2m}[/itex]-1)

The Attempt at a Solution


My main struggle here is with part A) ... C) I also am not 100% sure of.


A) I have managed to show dr/dt*=[itex]\frac{r-2m}{r+2m}[/itex] for the outgoing t*,r coordinates, by using null coordinates, but I am clearly missing something here.

B) *i have this one completed also*

C)I believe that the t coordinates give a truer physical description (for us), as these are introduced in order to adapt to the light rays. i.e. t* is the time coordinate adapted to the light rays, thus, this coordinate system should give the physical description for the light rays... but t should be physically truer for us (?)

D)I have said that all light rays near the r=2m will eventually hit the singularity, however the outward directed ray from outside the r=2m line will always remain outside r=2m, 'escaping' the black hole.

Any hints/help are greatly appreciated!
 
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  • #2


A) To show the equations for ingoing/outgoing radial light rays, we can start with the null condition for light rays in Schwarzschild spacetime: ds^2=0. Substituting in the Schwarzschild metric, we get:

0=-\left(1-\frac{2m}{r}\right)dt^2 + \left(1-\frac{2m}{r}\right)^{-1}dr^2 + r^2d\Omega^2

where d\Omega^2=d\theta^2 + sin^2\theta d\phi^2.

Next, we can use the substitution t_{*}=t+2mln(\frac{r}{2m}-1) to transform to the t*,r coordinates. This gives us:

0=-\left(1-\frac{2m}{r}\right)dt^{*2} + \left(1-\frac{2m}{r}\right)^{-1}dr^2 + r^2d\Omega^2

We can then solve for dt*/dr, which gives us:

dt*/dr = \pm\frac{r+2m}{r-2m}

This matches the equation given in the forum post for outgoing t*,r coordinates. For ingoing t*,r coordinates, we can use the substitution t_{*}=t-2mln(\frac{r}{2m}-1) to get the equation dt*/dr=-1.

B) The local light cones in t,r coordinates will look like standard Minkowski light cones, with the addition of a curvature term that causes the cones to tilt inwards towards the singularity at r=0. In t*,r coordinates, the light cones will be tilted inwards towards the singularity at r=2m due to the transformation.

C) The t,r coordinates give a truer physical description of the local light cones, as they are adapted to the curvature of spacetime caused by the presence of a massive object. The t* coordinate system is useful for studying the behavior of light rays, but it is not a physically realizable coordinate system.

D) As mentioned in the forum post, all light rays near the r=2m line will eventually hit the singularity. This is because the curvature of spacetime near the singularity becomes infinite, causing all geodesics (including light rays) to converge
 

1. What are Schwarzschild radial light rays?

Schwarzschild radial light rays are a type of light ray that travels in a radial direction in the spacetime surrounding a non-rotating spherical mass, as described by the Schwarzschild metric.

2. How are Schwarzschild radial light rays affected by gravity?

Gravity causes the path of Schwarzschild radial light rays to curve as they pass near a massive object. This is known as gravitational lensing and is a key prediction of Einstein's theory of general relativity.

3. Can Schwarzschild radial light rays escape from a black hole?

No, once a Schwarzschild radial light ray crosses the event horizon of a black hole, it can never escape. This is because the escape velocity at the event horizon is greater than the speed of light, making it impossible for any form of matter or energy to escape.

4. What is the significance of the radius of the photon sphere for Schwarzschild radial light rays?

The photon sphere is the region around a black hole where photons can orbit in stable circular paths. For Schwarzschild radial light rays, the radius of the photon sphere is equal to three times the Schwarzschild radius, and serves as a boundary between light rays that can escape from the black hole and those that are trapped within it.

5. How do gravitational waves affect Schwarzschild radial light rays?

Gravitational waves, which are ripples in the fabric of spacetime, can cause small perturbations in the paths of Schwarzschild radial light rays. This effect has been observed in the recent detection of gravitational waves from merging black holes, providing further evidence for the validity of Einstein's theory of general relativity.

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