# Photons and potential energy wells.

#### Atomos

Silly question:
Are photons traveling down into a potential (i.e. gravitational) energy well blue shifted, and ones traveling out red shifted? In other words, can a photon posses gravitational potential? If so why doesnt a derivation of the Schwarzschild radius using the concept of photon energetics yield the same expression?

given: E = hf, E = mc^2, Ep = -GMm/r

m = Ec^-2 = hf/c^2

hf = GM(hf)/(r * c^2)

1 = GM/(r * c^2)

r = GM/c^2

Rsch = 2GM/c^2

Isn't Rsch the radius of the body?

#### cesiumfrog

Yes, there is a (directly verified) colour shift. I would ascribe this more to time dilation than potential energy. If you want a silly $R_{schw.}$ derivation then the quickest way to satisfy you might be calculating the radius where classical projectile escape velocity is the speed of light. When you eventually learn real GR you should understand why the Schwarzschild solution needs to be more complicated, and in particular that its radius coordinate doesn't necessarilly mean what radius traditionally means.

#### pervect

Staff Emeritus
Atomos said:
Silly question:
Are photons traveling down into a potential (i.e. gravitational) energy well blue shifted, and ones traveling out red shifted?
Yes. Note that this is as measured by local clocks.

In other words, can a photon posses gravitational potential?
That's really a different question. It's possible in some circumstances, such as the static Schwarzschild metric, to define an "effective potential".

See for instance http://www.fourmilab.ch/gravitation/orbits/

In addition, you can use the Newtonian potential to approximate the redshift of a photon in a weak gravitational field, but this does not provide any insight into the gravitational field of a strongly gravitating body like a black hole, so the Newtonian potential is of no help in solving the strong field GR problem.

The conservation of energy doesn't provide any shortcuts to learning GR, in fact, learning how (and when) GR conserves energy is a rather advanced subject. For starters, see the sci.physics.faq

http://www.math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

Is Energy Conserved in General Relativity?

In special cases, yes. In general -- it depends on what you mean by "energy", and what you mean by "conserved".

#### pmb_phy

Atomos said:
Silly question:
Are photons traveling down into a potential (i.e. gravitational) energy well blue shifted, and ones traveling out red shifted?
Yes.
In other words, can a photon posses gravitational potential?
A photon can posses a gravitational potential energy yes. In the case of a Schwarzschild potential there is an explicit value which is given in an explicit relation. See the derivation at

http://www.geocities.com/physics_world/gr/red_shift.htm

Isn't Rsch the radius of the body?
No. The Schwarzschild radius is that "distance" Rsch from the center of the spherical body to the location of interest. Rsch is measured in terms the radius of the sphere that the point of interest lies on.

Pete

#### MeJennifer

Wait so gravity changes the frequency of photons? That is not my understanding of it at all.

Of course it is true that a photon that is measured in a different gravitational potential than the one it was emitted from shows a different frequency but the question is if we can attribute this to gravity.

It seems to me simply a relativistic effect. Last time I checked there was no absolute time and thus no absolute measure of frequency in GR.

Where am I wrong? pervect said:
Note that this is as measured by local clocks.
That does not make sense to me. A spaceship traveling from one gravitational potential to another would encounter a change in frequencies of light on board?

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#### Garth

Gold Member
MeJennifer said:
Wait so gravity changes the frequency of photons? That is not my understanding of it at all.

Of course it is true that a photon that is measured in a different gravitational potential than the one it was emitted from shows a different frequency but the question is if we can attribute this to gravity.

It seems to me simply a relativistic effect. Last time I checked there was no absolute time and thus no absolute measure of frequency in GR.

Where am I wrong? You are not wrong.

The photon is travelling along a null-geodesic, no forces are acting on it, no work is being done by it, it does not suffer gravitational potential energy changes. GPE is an inappropriate term to use in GR where free falling bodies experience no forces.

The change in frequency observed is a time dilation effect, where the time is measured and compared between clocks at different levels in the gravitational well. This time dilation is an effect of space-time curvature and as a consequence energy is not locally conserved in GR.

Garth

#### pervect

Staff Emeritus
Garth said:
You are not wrong.

The photon is travelling along a null-geodesic, no forces are acting on it, no work is being done by it, it does not suffer gravitational potential energy changes. GPE is an inappropriate term to use in GR where free falling bodies experience no forces.

The change in frequency observed is a time dilation effect, where the time is measured and compared between clocks at different levels in the gravitational well. This time dilation is an effect of space-time curvature and as a consequence energy is not locally conserved in GR.

Garth
I agree that gravitational potential energy is an inaproriate term to use in GR, except perhaps when one is doing a Newtonian or post-Newtonian approximation, and usually one then talks about the Newtonian potential to avoid confusion.

"Effective potential" and "energy at infinity" are OK (and have slightly different meanings, the last concept is probalby closest to what the OP had in mind).

I'm not sure that the O.P. has the background to appreciate these fine points, but I'm glad someone else objects to the idea of "gravitational potential" in GR.

Of course I still think it's simpler (and logically equivalent) to say that the photon redshifts as measured by local clocks. More on this in the next post.

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#### pervect

Staff Emeritus
MeJennifer said:
re: redshift as measured by local clocks:

That does not make sense to me. A spaceship traveling from one gravitational potential to another would encounter a change in frequencies of light on board?
The frequency or energy of a photon always depends on the frame of reference.

When one measures a free-falling photon with any clock, one has to implicitly or explictly set up a frame of reference for the measurement to make sense.

In this case, the frame of reference that I "set up" imagines observers that are "stationary", i.e. at constant Schwarzschild coordinates, and that these observers, using their local onboard clocks, each measure the frequency of a free-falling photon as it falls by them.
These observers get different numbers - they observe that the photons redshift.

I really don't see what's confusing about this. As measured by local clocks, the photons do redshift.

These observers also observe that their clocks tick at different rates given that one defines a way in which observers at different spatial locations compare clock rates. But to make this statement operationally meaningful takes work as well - one has to imagine that these observers are at constant Schwarzschild coordinates, as before, and that they rely on the fact that the geometry is static to insure constant travel times to carry out their sychronization procedures. Making these assumptions, the observers then find that their clocks tick at different rates, and that this is essentially the same effect as "gravitational redshift" under a different name.

#### Atomos

Thank you for your responses. I only have a high school background in physics, but thank you for providing me with interesting things to think about.

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