How would I go about finding the gravitational force produced by a photon? They are massless, but they must warp spacetime as all energy warps spacetime. What equations could I use to find the gravitational potential energy between two (very close together) photons in terms of their energy?
as i understand it (keep in mind that i am always wrong), a single photon has no defined location in spacetime between the time it is emitted and the time it is absorbed. without a location, it does not have a local effect on the gravitational field.
We don't have a quantum theory of gravity yet, so nobody can really answer your question. However, the Aichelburg-Sexl Ultraboost would be the spacetime produced by a very brief pulse of light.
Let's say you have a photon of energy E at a distance r from a material particle of mass M. Then the force between them can be found by using mass-energy equivalence, [itex]E=mc^2[/itex], giving [itex]F=GME/c^2r^2[/itex].
It doesn't seem like this should be accurate. If photons are truly massless, then I don't think you can just use E/c^2 as a replacement. Then again, I don't really know at all, which is why I asked. Do einstein's field equations work for energy in any form? Could they be used?
There is no reason why you could naively use Newton's law. You would have to find the stress energy tensor for a photon and solve Einstein's equations. But then things are tricky, since it's difficult to define a "gravitational potential energy" or a "force" in GR.
Most definitely not! You are correct, you cannot plug the energy into Newton's law to get the behavior of massless radiation. Einstein's field equations do apply for energy in any form, but Einstein's field equations also include terms for momentum, pressure, stress, and energy flux. When you are dealing with massless radiation the momentum terms are not negligible. The general class of spacetimes dealing with massless radiation is called pp-wave spacetime with the specific one for a brief pulse being the ultraboost that I linked to earlier.
Why not? I don't claim that it's perfectly accurate, but I think it will give roughly the right result. You have a stress-energy tensor for the photon. It has a time-time component which is equal to the energy density of the photon. It also has space-space components. It's true that the spacetime surrounding the photon won't be exactly a Schwarzschild solution (due to the space-space part of the stress-energy tensor). But you will certainly get approximately the right answer this way. One way to see that it can't be wildly wrong is that the equivalence principle tells us that the real and virtual photons inside a piece of matter have gravitational fields exactly like the gravitational fields contributed by the rest mass of the material particles. In matter, the space-space part of the stress-energy tensor averages out to zero by symmetry. The lack of this cancellation for a single photon will change the result somewhat, but not by an order of magnitude. If the photon was bouncing back and forth between two mirrors, then the e.p. argument would become exact, and the photon's gravitational force would be given by exactly the expression I gave earlier. Yet another way to see that it can't be wildly wrong is that there is simply no other way to put together the variables in the problem to give an answer with the right units and the right dependence on the variables (e.g., proportionality to E to the first power). And I would claim that this would give the same result as the equation I gave, to a reasonable approximation. The lack of a quantum theory of gravity is completely irrelevant here. It would only be relevant if there were some scale in the problem that was at the Planck scale.
[itex]E = mc^2[/itex] does not apply to massless particles. You cannot justifiably use it to get your result, even if it somehow happened to be correct. It may be approximately the correct result on dimensional grounds, but that's not a good enough reason to use [itex]E = mc^2[/itex] for the derivation when it's not correct.
I already explained why not, the momentum terms are not negligible. Newton's formula is an approximation to GR only in a specific limit, namely the limit as M->0 and the limit as v<<c. Obviously for a photon v=c so the approximation is simply not valid.
Hi bcrowell, I'm with DaleSpam here, the Photon's gravity seems to be more like a Dirac-pulse. I mean, maybe you can get to a similar result if you include the finite propagation speed of gravity, and model some "sonic bang". But that's far from the formula you gave. No, that's not right. The stress-energy tensor still includes pressure, wich is a space-space component (only the time-space components vanish). And pressure contributes to gravity: the source term for a perfect fluid like this is the trace of T, not the tt-component. Thus, a photon gas has double gravitation compared to pressureless dust of the same mass. To get some closed system to push around and test the EP, you'll have to keep the pressure in some containment, which experiences tension and so cancels the effect of pressure.
excuse me, but let me repeat something i stated above. a single photon does not a defined location in spacetime. with no location, it cannot have a gravitational effect. if i am wrong about that, i really need someone to clarify for me. thanks.
It has a defined location. You can't define a frame in which the photon is at rest, but in every other frame it has a well defined position at each time.
The problem is, we do not currently have any theory that combines general relativistic effects with quantum effects, so the question for a single isolated photon, or a pair, can't be answered from a quantum theoretical point of view. (In quantum theory a photon's location cannot be measured with perfect precision. In fact, if you don't measure it, it can be in several places at once.) So the best we can do is consider a "classical photon" (if that's not a contradiction) which behaves as a non-quantum particle. The question is less problematic (from the quantum view) if we consider instead a collection of, say, a trillion photons whose collective behaviour can be modelled more accurately.
There have been candidates put forward (by Margaret Hawton and others), but I do not think there is an accepted position operator for the photon in quantum theory.
Are you referring to the fact that there is uncertainty in the photon's location? That is true of anything, including the Earth (which definitely has a gravitational effect.)
If matter annihilates with antimatter, the created photon gas *MUST* has exactly the same gravity as matter before, otherwise GR would be violated. Of course, rest mass is zero, so gravity is created by the pressure components of the tensor.