I have a few questions (sort of on topic but not much related to BH.):
First is: Do photons have a very tiny gravitational field?
They have no rest mass so I guess the answer is no* (and at end of post support that with simple observation). Here is the problem that makes me have some doubt:
An electron and positron far from all other mater and 2 cm from each other, I think, could be orbiting about their mass center. A meter from it I think there should be a weak and slightly modulated gravitational field. (Modulated as in 1/4 the orbit period the field at 1M from the center of mass changes in magnitude only (not direction) from:
M/(1.01)^2 + M/(0.99)^2 To 2M/(1.0...0X)^2
(Yes I'm too lazy to find X and how many zeros between X and the 1.0 as I'm just justifying "modulated")
Second question: Does this modulation at point 2 meters form the mass center have speed of light delay? - I. e. does not have its max strength (about 25% as strong) exactly when field at 1M from mass center does have its max? - I think "yes.
Third question: These two charges are accelerating so are radiating. Hence spiral closer together will they not? Again, I think "yes."
Fourth question: Eventually they will transform into two ~ 511Mev gamma rays, that travel away from the old mass center point (perhaps the still mass center point if answer to question 1 was yes) will they not?
Fifth question: Does the slightly modulated gravitational field speeding off into space continue but now as a growing "no field hole" centered on the old mass center point? If answer to Q1 was yes, I think there is no hole as that remains forever the mass center of the two gamma rays, but if answer to Q1 is no then I guess there is an expanding field free hole.
Now for my observation supporting the No answer to Q1:
On a clear night I look at some stars - first with one eye then with the other. It is fact that both eye see the same star field. But consider the small (incredibly small) angle the path of the photons reaching my right eye differs from the path of the photons coming to my left eye. If these photons did have mutual gravitational attraction for say 100,000 light year long paths, then the "photo light field" should be "quills of light" with no light voids between. I. e. the field of stars seen by my right eye should not be the same as that seen by my left eye. Likewise, If I did not like the star field one eye was seeing, (other closed) just move head a little to see a different star field.
Comments (or at least what answers do you think correct to my 5 questions?)
* Also the photon's M = E/(c^2) mass is "frame dependent" so that would seem to be a problem with their "bending of space" view of gravity, I think. I do lean to the POV that all energy which is the same in all frames does make gravity. I.e. the gravity from a hot brick deceases slightly as it cools - Its temperature (random KE) is same in all frames. Any arguments on this POV? Note that jar of half ice & half water has T = 0C in all frames.
Billy, you present a series of questions that are very difficult to answer from a single theoretical basis. It seems your questions assume an inherent definition of gravity consistent with GR, and yet most of your questions/thought experiment are formed within the context of an interaction between fundamental particles, which is the domain of QM and an area where the charge relationship dominates the interactions. Gravity has not yet been successfully incorporated or explained within the context of QM. Still...
To the issue of whether a photon has a gravitational field, I would say no.., at least not directly and/or independently. But even this depends on the theoretical context you approach the issue from. If you approach that question from some interpretations of GR all energy contributes to a gravitational field. Some would interpret this to mean that even the photon has (or contributes to) a gravitational field. If on the other hand you begin from some of the theoretical work of Haisch, Rueda & Puthoff dealing with inertia and gravitation as emergent from an interaction between vacuum energy (specifically an EM spectrum of the ZPF) and charged particles (which could be extrapolated to matter), photons would have no independent gravitational field, while they would play a role in how inertia and gravitation emerges from quantum phenomena. (Note that Haisch et al are only dealing with the issue as boundary conditions between charged fundamental particles and an isotropic EM ZPF background. Extending their theoretical work to a classical scale is entirely speculative on my part... But it would suggest that the photons associated with the EM ZPF spectrum do not independently gravitate.)
I have a lot of trouble addressing the questions that follow, as individual questions, because they seem to me to be questions which would be dominated by QM rather than GR and gravity (Even though my comment above does just that.., crosses that divide). As example, when you begin with an orbiting electron/positron pair, even if those particles generate any independent gravitational field, they are point particles and their charge interaction would be far greater than any possible gravitational field.
While gravitation seems to have a far greater reach than, charge related interactions, it is also generally a far weaker force. It is only the extreme gravitational forces found at the centers of stars and black holes where gravity overcomes those charge related forces associated with atomic stability. This would not be the case when dealing with individual fundamental particles.
It does not seem to me that gravity can logically be thought of as playing a role in the hypotheticals you present.
Your last paragraph brought to mind a quote, which I am sad to say, I cannot find the reference for right now. It went something like this,
while it is true that all mas can be thought of as energy (or an expression of energy), not all energy is (or can be thought of), as mass.
Keeping all of the above in mind, it seems that your thought experiment is attempting to ask what happens to the gravitational field if the source, the electron/positron pair in your example, annihilate one another radiating away as two gamma rays. To that my answer would be that the gravitational field dissipates at the speed of light outward from the center of mass of the initial electron/positron pair. The mass has been converted completely to a non-mass form of energy, the gamma rays. From the Haisch et al perspective, the gamma photons would have no independent gravity, so the initial field would just dissipate at the speed of light.
If the gamma photons, do have independent gravitational fields, the change in the combined field would still occur at the speed of light, photons propagate at the speed of light, but the change in the gravitational field would be the same as if you were dealing with two stars, initially orbiting one another.., and then flying off away from their initial center of mass. The only difference would be how fast the change happens.
If this last accurately describes your overall question, it really did take me all of the above rambling to get there.
