Computing Null Geodesics in Schwarzschild Geometry

In summary, computing timelike geodesics in the Schwarzschild geometry involves using conserved quantities and an effective Lagrangian to determine the geodesics. When computing a null geodesic, the biggest change is that proper time cannot be used as the parameter and the 1 in the equation for conservation of the t component of momentum becomes a zero. The conserved energy and angular momentum quantities take the same form as for timelike geodesics, but the parameter lambda does not have a direct physical meaning and can be reparametrized without changing the tangent vector.
  • #1
stevendaryl
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Computing timelike geodesics in the Schwarzschild geometry is pretty straightforward using conserved quantities. You can treat the problem as a variational problem with an effective Lagrangian of

##\mathcal{L} = \frac{1}{2} (Q \frac{dt}{d\tau}^2 - \frac{1}{Q} \frac{dr}{d\tau}^2 - r^2 (\frac{d\theta}{d\tau}^2 + sin^2(\theta) \frac{d\phi}{d\tau}^2))##

where ##Q = 1 - \frac{2GM}{r}##

This "lagrangian" leads to the following conserved quantities:

  1. ##K = Q \frac{dt}{d\tau}##
  2. ##L = r^2 \frac{d\phi}{dt}## (You can choose ##\theta## and ##\phi## so that ##\theta = \frac{\pi}{2}##, so all the radial motion is due to changing of ##\phi##
  3. ##\mathcal{L}## itself, which is always equal to 1/2.

In terms of these conserved quantities, the geodesics are completely determined by ##\frac{dr}{d\tau}##, which satisfies the one-D equation:

##\frac{1}{Q} (K^2 - \frac{dr}{dt}^2 ) - \frac{L^2}{r^2} = 1##

My question is: How are things changed if we are computing a null geodesic, instead of a timelike geodesic? The biggest change is that you can't use proper time as the parameter (since it's identically zero for null geodesics, by definition). If you replace ##\tau## by a different parameter, ##\lambda##, I'm assuming that it's still true that there is something like angular momentum that is conserved, but I'm not sure about the first equation, which is about the conservation of the ##t## component of momentum.
 
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  • #2
The 1 becomes a zero, I think.

Carroll does it by observing that there is a Killing field parallel to ##\partial_t## and another parallel to ##\partial_\phi##, hence writing expressions for ##dt/d\lambda## and ##d\phi/d\lambda## in terms of conserved quantities along a geodesic, restricting himself to the equatorial plane, noting that ##g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}## is either 0, -1, or +1, and solving for ##dr/d\lambda##. I think that's your last equation, give or take the value on the RHS.
 
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  • #3
Indeed, the 1 becoming a zero is the only change. The conserved ”energy” and ”angular momentum” quantities take the same form as they are the inner product between an affinely parametrized geodesic’s tangent and a Killing field.
 
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  • #4
Orodruin said:
Indeed, the 1 becoming a zero is the only change. The conserved ”energy” and ”angular momentum” quantities take the same form as they are the inner product between an affinely parametrized geodesic’s tangent and a Killing field.
So what is the parameter ##\lambda## for a null geodesic?
 
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  • #5
stevendaryl said:
So what is the parameter ##\lambda## for a null geodesic?
It does not have direct physical meaning the same way proper time does as you can reparametrize the geodesic without changing that the tangent vector us null.

If you want you could choose it such that the tangent vector is the 4-frequency of a light signal following that geodesic. However, there can be several such signals following the same geodesic.
 
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Related to Computing Null Geodesics in Schwarzschild Geometry

1. What is the significance of computing null geodesics in Schwarzschild geometry?

Computing null geodesics in Schwarzschild geometry allows us to understand the paths of light rays in the presence of a massive object, such as a black hole. This is important in understanding the effects of gravity on light and can help us make predictions about observations of light from distant objects.

2. How is the Schwarzschild geometry related to general relativity?

The Schwarzschild geometry is a solution to Einstein's field equations in general relativity. It describes the curvature of spacetime around a non-rotating, spherically symmetric mass. Computing null geodesics in this geometry helps us understand how the presence of mass affects the curvature of spacetime.

3. What is the mathematical process for computing null geodesics in Schwarzschild geometry?

The mathematical process involves solving the geodesic equation, which describes the path of a free-falling particle in curved spacetime. In the case of null geodesics, the particle has zero mass and travels at the speed of light. The equation involves solving for the time and spatial coordinates of the particle as it moves through the curved spacetime.

4. How does the curvature of spacetime affect null geodesics in Schwarzschild geometry?

The curvature of spacetime is directly related to the presence of mass in Schwarzschild geometry. As the mass increases, the curvature of spacetime becomes more pronounced, resulting in a greater bending of light rays. This means that the paths of null geodesics will be more curved and may even result in light being trapped in a circular orbit around the mass (known as a photon sphere).

5. What applications does computing null geodesics in Schwarzschild geometry have?

Aside from helping us understand the behavior of light in the presence of massive objects, computing null geodesics in Schwarzschild geometry has practical applications in fields such as astronomy and astrophysics. It can be used to make predictions about the motion of stars and galaxies in the vicinity of massive objects, as well as to interpret observations of gravitational lensing.

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