Is Gravitational Potential Energy Not Really Energy for Photons?

[PhiJ]

By reletivity,
K.E.=mc^2-m₀c^2
For light, m₀=0, so K.E.=mc^2

We know that photons have gravitational potential energy, but that all of the photon's mass-energy is kinetic energy, so doesn't that mean that gravitational potential energy is not really energy, and thus the mass energy of the universe is not constant?

There must be something wrong with that argument, but I can't see it.

 

[Garth]

Photons do not have gravitational potential energy - that concept is foreign to GR, which conserves energy-momentum not, in general, energy .

For a photon $$E = h\nu$$

and energy is a frame dependent quantity, a particle's energy is measured as

$$E = -p_{\alpha}U^{\alpha}$$

where $$p_{\alpha}$$ is the particle's 4-momentum and $$U^{\alpha}$$ is the observer's 4-velocity.

Garth

 

[PhiJ]

Oh! I knew it would be obvious!

 

[pmb_phy]

By reletivity,
K.E.=mc^2-m₀c^2
For light, m₀=0, so K.E.=mc^2
We know that photons have gravitational potential energy, but that all of the photon's mass-energy is kinetic energy, so doesn't that mean that gravitational potential energy is not really energy,...

I can't see how you came to that conclusion since you used an expression from SR to make a conclusion in GR. E.g. In the weak field limit the total energy for a partilce in a gravitational field is the rest energy + kinetic energy + potential energy. If the rest energy is zero then what is left is kinetic energy + potential energy. You can see this derivation on my website here

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

The old definition of potential energy in my opinion should have been replaced by a new one which can be stated as "Energy of a particle by virtue of position only." There are complications with that since its bound to be misused.

Pete

 

[Rebel]

about..

I have another energy related question:
If the universe is expanding in a way explained at least in part by a posittive cosmological constant, even then one assumes the energy of the universe to be constant? or is it explained using other therms? I was wondering about it cause there are some hyphotesis where universe is accelerating in some stages and in others is decelerating, some using tachyons to explain the dark energy. I believe this should be a though work in relationing cosmology with thermodynamics, e.g. how fundamental are the laws of thermodynamics compared with others.

 

[pervect]

Take a look at the sci.physics.faq on Energy & Gr.http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html"

The Cosmic Background Radiation (CBR) has red-shifted over billions of years. Each photon gets redder and redder. What happens to this energy? Cosmologists model the expanding universe with Friedmann-Robertson-Walker (FRW) spacetimes. (The familiar "expanding balloon speckled with galaxies" belongs to this class of models.) The FRW spacetimes are neither static nor asymptotically flat. Those who harbor no qualms about pseudo-tensors will say that radiant energy becomes gravitational energy. Others will say that the energy is simply lost.

The textbooks I've seen, such as MTW and Wald, make a point of saying that the universe doesn't have a well-defined energy.

 

[Rebel]

T

Thank you pervect. Actually I am reading the MTW book, but it will take me a lot more of time to finish it. Regards.

 

[pervect]

LOL, I'll bet it will take just a little bit of time to read all 1200+ pages.

Since you have MTW, you might want to check out pg 457 "Mass and angular momentum of a closed universe" in MTW, also pg 705 has some useful notes on the topic.

 

[PhiJ]

OK, there seems to be a disagreement.
Can anyone back pmb phy or Garth up, and pmb phy, if light does have GPE, won't this then add to it's mass and then slow it down?

 

[Garth]

OK, there seems to be a disagreement.

Where's the disagreement? (So far)

Can anyone back pmb phy or Garth up, and pmb phy, if light does have GPE, won't this then add to it's mass and then slow it down?

That is a good reason why it doesn't have GPE! As I said GPE is a classical concept that does not carry through to GR. In GR gravitational red-shift is the observation of a time dilation effect between the bottom and top of a gravitational 'pit'.

The null-geodsics of consecutive pulses of light diverge as they traverse curved space-time and are received further apart in time at the top of the pit than they were transmitted at the bottom.

If you do have access to Misner Thorne and Wheeler - Gravitation - you need to read 7.3 page 187 - 189 carefully. They follow an argument of Schild which uses the diagram fig 7.1 to prove that if space-time were flat (SR) the null-geodesics of succesive pulses of light ascending a gravitaiton 'pit' will not diverge and red shift will not be observed.

To understand what really is going on you have to draw the diagram on a curved surface, the inside surface of the Schwarzschild 'funnel', to make the null-geodesics diverge.

I hope this helps.

Garth

 

[pervect]

Other than my previously referenced FAQ article,

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

you'll have to read some textbooks if you want more info. MTW's gravitation is good, though somewhat dated. Wald's "General Relativity" has one of the better modern treatments of energy in GR, Be prepared for some heavy reading, though - especially in Wald, MTW has an informal style that might allow someone to get some understanding without the math, Wald's approach is rather difficult.

You might also look at the quote from Steve Carlip that I posted in another thread

https://www.physicsforums.com/showpost.php?p=810720&postcount=4

Here's a link to some of the honors Steve Carlip has won

http://www.physics.ucdavis.edu/Text/Carlip.html#Honors

While this doesn't necessarily mean he can't be wrong, he's definitely a heavy hitter in the relativity world, and one of the few such who makes some "outreach" efforts to talk about GR.

 

[PhiJ]

pmb phy said:

In the weak field limit the total energy for a partilce in a gravitational field is the rest energy + kinetic energy + potential energy

so he's saying that there is GPE in GR, and your saying their isn't.

Unless I'm just misinterpreting or generally being stupid...

 

[pmb_phy]

...and pmb phy, if light does have GPE, won't this then add to it's mass and then slow it down?

The gravitational potential energy of position will not add to the photons rest mass and make it non-zero. It will, however, add to its "relativistic" mass and alter it, yes. So in that sense it does change its mass, just not its proper mass. It is also a matter of observation/prediction etc. the coordinate speed of light in a gravitational field changes.

Note: The increase in relativistic mass is not the reason for the slowing of light in a g-field. The speed of anything in a g-field is independant of its mass. In this case the inertial mass of light changes along with its passive gravitational mass so as to cancel out.

Pete

 

[pmb_phy]

They follow an argument of Schild ...

The argument of Schild's was argued to be wrong in the American Journal of Physics. I disagree with Schild myself and have explained myself in that link I gave above as I recall.

Pete

 

[pervect]

OK, there seems to be a disagreement.
Can anyone back pmb phy or Garth up, and pmb phy, if light does have GPE, won't this then add to it's mass and then slow it down?

You appear to be conflating invariant mass with "relativistic mass".

"Slowing it down" refers to the fact that only objects with an invariant mass of zero can travel at the speed of light. Photons always have a zero invariant mass in both GR and SR as I will explain below.

"Adding to its mass" refers to the "relativitic mass", which in the case of a photon is just another name for its energy.

In SR, invariant mass is given by E^2 - px^2 - py^2 - pz^2, and is always a constant, zero for a photon. E is the energy, and px, py, and pz are the x, y, and z components of the momentum.

In GR invariant mass is given by a different formula:

g_00 E^2 - g_11 px^2 - g_22 py^2 - g_33 pz^2

in a diagonal metric (not the most general possible, but easy to write down and understand). For completness, in the most general case

$$m = g_ij P^i P^j$$

where P^i is the energy-momentum 4-vector (P^0 being the energy).

As is the case in SR, in GR the invariant mass of a photon is always zero.

Note that an effective gravitational potential can exist in very simple cases in GR (such as a photon or particle falling into a black hole). However, the general existence of such an effective potential is not guaranteed, it exists only when the system is static.

Note also the use of the word "effective".

 

[pmb_phy]

For completness, in the most general case
$$m = g_ij P^i P^j$$

To be exact that should read

$$m^2 = g_{\alpha\beta} P^{\alpha}P^{\beta}$$

You forgot the square on m pervect (I like to use greek letters for indices which range from 0->3).

Pete

 

[PhiJ]

Arite, I seem to be sort of understanding you, and it looks like the rest I will understand after I read my differential geometry + GR text.

Does the space-time contain the GPE then?

btw. How do you do LaTeX images on these forums?

 

[selfAdjoint]

btw. How do you do LaTeX images on these forums?

You enter [ tex ] -your latex code- [ /tex ], without the spaces in the brackets.