Understanding the Gravitational Redshift Caused by a Black Hole

[Karmyogi01]

TL;DR

The theory of gravitational redshift caused by a blackhole and its implications. Conservation of energy as frequency of radiation changes during its transit through a strong gravitational field.

 

[Bosko]

I suggest you start with the formula for wavelength for an observer at infinity.

$$\frac { \lambda_\infty} {\lambda_e} = \sqrt { 1- \frac {r_s} {r_e} } $$
where $r_s$ is the Schwarzschild radius

 

[Ibix]

TL;DR Summary: The theory of gravitational redshift caused by a blackhole and its implications. Conservation of energy as frequency of radiation changes during its transit through a strong gravitational field.

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Please don't post images of text and maths. It's pretty near illegible, especially in dark mode, and we have LaTeX here.

You are considering the case where $r_o$ and $r_e$ are both less than $r_s$. What do you know about the $r$ coordinate inside a black hole? And what does $r_e<r_o$ imply in that case?

 

[PeterDonis]

@Karmyogi01 your expression for the redshift is meaningless at and inside the horizon. It is only valid for radiation that goes between hovering observers, i.e., observers who maintain the same $r$ coordinate, and there are no such observers at or inside the horizon.

 

[PeterDonis]

I suggest you start with the formula for wavelength for an observer at infinity.

$$\frac { \lambda_\infty} {\lambda_e} = \sqrt { 1- \frac {r_s} {r_e} } $$
where $r_s$ is the Schwarzschild radius

He already did that; his formula is just the ratio of the factor you give for two observers at different $r$ coordinates.

 

[Orodruin]

@Karmyogi01 your expression for the redshift is meaningless at and inside the horizon. It is only valid for radiation that goes between hovering observers, i.e., observers who maintain the same $r$ coordinate, and there are no such observers at or inside the horizon.

Maintain the same $r$ coordinate and angular coordinates (in standard Schwarzschild coordinates). In other words, observers whose world lines are the integral curves of the time-like Killing field $\partial_t$.

 

[Karmyogi01]

@Karmyogi01 your expression for the redshift is meaningless at and inside the horizon. It is only valid for radiation that goes between hovering observers, i.e., observers who maintain the same $r$ coordinate, and there are no such observers at or inside the horizon.

Thanks for all the very helpful insights provided here! I truly appreciate it! So, if this expression for the redshift is not valid for this region under the blackhole boundary, which one is the right expression? Let us consider a scenario wherein the mass of the blackhole increases, when radiation was in progress from source (emitter) to observer, both being originally outside the blackhole boundary. As the mass of the blackhole increases (may be due to gobbling up of some object nearby), and suddenly both of these points fall within the blackhole boundary. What happens to the original radiation that was in progress outward from the source to the observer/detector. Any insight into it is greatly appreciated.
TL;DR Summary: The theory of gravitational redshift caused by a blackhole and its implications. Conservation of energy as frequency of radiation changes during its transit through a strong gravitational field.

View attachment 362489

 

[PAllen]

Since there is no special class of hovering observers, you need to take account of the motion of the emitter and receiver. Then the formula would then be more complex with more variables. One special class of observers to consider would be so called raindrop trajectories - free fall from infinity. For computational ease, given such observers, you might use Gullestrand-Panieve coordinates or Lemaitre coordinates, as both are based on such raindrop observers.

 

[PAllen]

Your scenario of a growing BH is needlessly complex. Consider, instead, two raindrop observers, one just ahead of the other. The outer one sends a signal to the inner one from outside the horizon, that will reach the inner one inside the horizon. Or you could have the outer one one send the signal from inside the horizon, such that it will reach the inner one before the inner one reaches the singularity.

 

[PeterDonis]

if this expression for the redshift is not valid for this region under the blackhole boundary

It's even more restricted than that; both observers have to be outside the horizon and stationary, i.e., unchanging $r$, $\theta$, and $\phi$ coordinates.

which one is the right expression?

There is no single general expression for the redshift between two observers. It depends on where they are and what their state of motion is. In some cases there is no way for one observer to send light signals to the other at all--for example an observer inside the horizon can't send light signals to an observer outside.