Gravitational Redshift and its effects on Photon

[Arman777]

Let us assume that we have a large gravitational field, then the gravitational redshift can be expressed as,

$$\frac {v_{\infty}} {v_e} = (1-r_s/R_e)^{1/2}$$

In this equation $v_{\infty}$ represents the frequency of the light measured by an observer at infinity, $v_e$ is the frequency of the emitted wavelength, $r_s$ is the schwarzschild radius, $r_S=2GM/c^2$, and finally $R_e$ is the radius which photon is emitted.

And I argue that when we $R_e$ get closer to $r_s$ the wavelength of photon will decrease.
And at $r_s=R_e$ the energy of the photon will be zero, at the surface of the event horizon
"When the photon is emitted at a distance equal to the Schwarzschild radius, the redshift will be infinitely large, and it will not escape to any finite distance from the Schwarzschild sphere."
https://en.wikipedia.org/wiki/Gravi..._equivalence_principle_and_general_relativity
At this point I argue that the energy of the photon also decreases. I find an experiment that explains the energy decrease, https://en.wikipedia.org/wiki/Pound–Rebka_experiment and the math of it
http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/gratim.html

At this point, can someone else argue that "Light doesn't lose energy as it ascends. It was emitted with less energy at a lower elevation." ? I didnt understand what it means and also how it explains the experiment that I shared. Thanks

 

[Matterwave]

Can you explain where the quote "Light doesn't lose energy as it ascends. It was emitted with less energy at a lower elevation." comes from? Is that your own hypothesis, or did you read it in a book or lecture somewhere? On the face of it, that statement appears to be simply false - but I might be missing some context.

 

[Arman777]

Can you explain where the quote "Light doesn't lose energy as it ascends. It was emitted with less energy at a lower elevation." comes from? Is that your own hypothesis, or did you read it in a book or lecture somewhere? On the face of it, that statement appears to be simply false - but I might be missing some context.

A guy that I am arguing with. I don't know why he claims such thing. he said like this, "but the energy of the photon doesn't decrease. Take a look at page 149 of Relativity, the Special and General Theory. Einstein said “an atom absorbs or emits light at a frequency which is dependent on the potential of the gravitational field in which it is situated". When the ascending photon ascends, its E=hf energy does not reduce, and nor does its frequency. There is no outflow of energy from the photon. Instead the photon was emitted at a lower frequency at a lower elevation, with less energy"

 

[Dale]

"Light doesn't lose energy as it ascends. It was emitted with less energy at a lower elevation." ?

This is a tricky statement. There is a way to interpret it that is wrong and a way that is right.

In a coordinate independent sense the light travels on a geodesic. Because it is on a geodesic there is no point on its path where any four-momentum was added or removed from the light. Energy is the timelike part of the four momentum so no loss of four momentum means no loss of energy.

However, in a coordinate dependent sense, in Schwarzschild coordinates the timelike component of the four momentum does decrease. The formula that you posted is an example of a coordinate-based equation specific to Schwarzschild coordinates that shows this decrease in energy. However, because it is a coordinate based equation it only applies specifically when using Schwarzschild coordinates and can be eliminated or changes in other coordinate systems.

 

[Arman777]

This is a tricky statement. There is a way to interpret it that is wrong and a way that is right.

In a coordinate independent sense the light travels on a geodesic. Because it is on a geodesic there is no point on its path where any four-momentum was added or removed from the light. Energy is the timelike part of the four momentum so no loss of four momentum means no loss of energy.

However, in a coordinate dependent sense, in Schwarzschild coordinates the timelike component of the four momentum does decrease. The formula that you posted is an example of a coordinate-based equation specific to Schwarzschild coordinates that shows this decrease in energy. However, because it is a coordinate based equation it only applies specifically when using Schwarzschild coordinates and can be eliminated or changes in other coordinate systems.

So only in Schwarzschild coordinate system energy decreases and in others don't ?

 

[Dale]

So only in Schwarzschild coordinate system energy decreases and in others don't ?

I am sure that there are other coordinate systems where the energy decreases besides Schearzschild, but it is not true in all coordinate systems. If desired you could even construct coordinate systems where the energy increases.

 

[Arman777]

I am sure that there are other coordinate systems where the energy decreases besides Schearzschild, but it is not true in all coordinate systems. If desired you could even construct coordinate systems where the energy increases.

Its strange, but also makes sense at some point. So I am true but also he is true ..
So the bottom line is, Energy is coordinate dependent .. However can we argue that "the high redshift at the event horizon is the reason why light cannot escape from a black hole" ?

 

[Orodruin]

Thinking of energy as an inherent property of a null geodesic is not very useful unless you make it the constant of motion related to invariance of the Schwarzschild metric under time translation. That by definition is a constant of motion. If you think about frequencies, there is no such thing as the frequency of a given photon. In order to ascribe a frequency you need to define an observer and the observed frequency becomes a function of the observer-photon system.

 

[Arman777]

I posted the threat as beginner, Its really hard for me to understand what's going on.

 

[Dale]

However can we argue that "the high redshift at the event horizon is the reason why light cannot escape from a black hole" ?

You could argue that, as long as you realize that is not the only reason. Many people will prefer other valid reasons.

 

[Arman777]

You could argue that, as long as you realize that is not the only reason. Many people will prefer other valid reasons.

What are the other reasons ?

 

[Nugatory]

Its strange, but also makes sense at some point. So I am true but also he is true ..
So the bottom line is, Energy is coordinate dependent ..

Energy is coordinate dependent, and so is redshift and blueshift. Frequency is defined as the number of wave crests that come by in a second, so clearly frequency and redshift/blueshift will be affected by your frame-dependent notion of what a second is.

However can we argue that "the high redshift at the event horizon is the reason why light cannot escape from a black hole" ?

We can, but it's probably not the most helpful way of explaining it. For example, that explanation would confuse someone who is free-falling into the black hole because using their frame there is no redshift at the horizon.

Perhaps the most straightforward explanation is the coordinate-independent one: There are no light-like paths (this is, the straight-line paths that a flash of light will follow) that lead out of the black hole from the horizon.

This all becomes perfectly clear in a Kruskal diagram.

 

[Dale]

What are the other reasons ?

There are a lot of reasons you could give. My preferred reason would be that all outgoing paths are spacelike at and beyond the horizon. Another explanation is that the horizon itself is a null surface. I prefer those since they are both local and coordinate independent.

 

[PeterDonis]

the gravitational redshift

This is not "the gravitational redshift" without qualification. To be precise, it is the gravitational redshift of light emitted by an object "hovering" at rest at a given altitude, when that light is received by an observer at rest at infinity.

With that more precise definition, it should be evident why the formula you give does not apply at or inside the horizon: because it is impossible for anything to "hover" at those altitudes, since doing so would require such an object to have a worldline that was not timelike (null at the horizon, spacelike inside the horizon), which is impossible.

So the bottom line is, Energy is coordinate dependent .. However can we argue that "the high redshift at the event horizon is the reason why light cannot escape from a black hole" ?

As you might be realizing, this is probably not the best way to think of it. I think a better way is to recognize, as @Orodruin pointed out, that a light ray does not have an "energy" or a "frequency" in isolation. It has a definite frequency (and hence energy) when emitted, and it has a definite frequency (and energy) when received. But what those definite frequencies are depends on the specific worldlines of the emitter and receiver and the worldline the light ray takes between them. In the "gravitational redshift" formula you wrote down, all of those things are fixed to particular values (the two objects are at rest relative to the hole's horizon, at the given altitudes, and the light ray goes radially outward from one to the other). A significant limitation of the Wikipedia article you reference, and many other pop science articles on the topic, is that they fail to make the crucial distinction between this specific scenario, where all of the values are specific, and the generic scenario in which they are not.

 

[jartsa]

At this point, can someone else argue that "Light doesn't lose energy as it ascends. It was emitted with less energy at a lower elevation." ?

Well let me try to show how that could make sense:

Let's say we have a flashlight with mass 1 kg and a gravitating body with Komar mass of 10 solar masses.

Now we lower the flashlight on to the surface of the gravitating body, this causes the flashlight to become time dilated, let's say by 50%. The flashlight is now part of the gravitating object, and the Komar mass of the gravitating object has increased by 0.5 kg or less. (To get Komar mass we multiply mass at infinity by the time dilation factor)

Ok now the flashlight gets fully converted to energy, which is beamed up from the gravity well and then converted to matter. There will be 0.5 kg of said matter. And the mass of the gravitating body is its original mass.

So we could say 0.5 kg of Komar mass was removed from the object and 0.5 kg of Komar mass arrived at much higher position.

(And then we convert to matter that energy that we got when we were lowering to the flashlight - there will be 0.5 kg of that matter)

 

[PeterDonis]

Instead the photon was emitted at a lower frequency at a lower elevation

This is obviously false since the energy and frequency of the photon when it's emitted can be measured, and when they are, they are not lowered.