Genady

Schwarzschild metric and all related derivations such as photon sphere, are valid in vacuum. If we start adding a significant amount of energy there, we need to consider a different metric and different effects.

Pick any: Calculate Pi with Python - GeeksforGeeks

There are many formulas and algorithms for calculating \(\pi\). See, e.g., here: Approximations of π - Wikipedia

Let "result" to be a b's digit of the number \(\pi\). I don't know it beforehand, but they will be equal.

So, what is wrong with defining, say, result1==result2==5?

I remember that charged particles moving in rotating magnetic fields are involved in the description of this, but it was too long ago (and not in my main line of study) to recall details. Hope to see some expert answers.

Philosophy is certainly neither pseudoscience nor science. It is also not football, chess, music, engineering, cooking, etc. I think that "love of wisdom" is a good definition.

Well, the instantaneous change of acceleration, in the case of circular motion, happens to be also "along the direction of motion, perpendicular to the instantaneous acceleration". However, I don't know how they (i.e., the time derivative of acceleration and the radiation) are related and don't claim anything in this regard. My point in this post was that AFAIK, a constant acceleration of charged particle can be insufficient to cause radiation.

The direction of the acceleration changes. The vector rotates.

Something is missing in this definition. As it is described, the simple solution would be just to have a "result" some constant or, more generally, independent on the variable "a".

It is not. It is an example of not every acceleration resulting in radiation, i.e., in these examples a changing acceleration results in radiation.

AFAIK, not every acceleration causes radiation from a charged particle. For example, in synchrotron radiation the acceleration is perpendicular to the particle velocity. In some other cases, a magnitude of acceleration is variable. Is it correct?

Are they? Don't they collide with each other and affect each other non-gravitationally?

If the light orbiting the BH is not a test particle but rather has the energy as described in the OP, then I think it is not on a photon sphere anymore. The "photon sphere" of the original black hole would be inside the new black hole.

Is there a stable circular orbit for a massless test particle around a Schwarzschild black hole? Photon sphere is unstable, AFAIK.

You are free to guess.

45 years ago President Carter has helped me to escape from the USSR.

Moreover, it is not necessary for c to be a speed of anything. It is a coefficient in the spacetime metric.

I am sorry, I didn't pay attention and did not notice that the question was directed to you. Deleted.

Another way to look at it is that mass of an affected body was introduced into the equation to make it fit into the force-based model. Then, it disappears when the model is not force-based anymore.

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In QFT, photon does not have dimensions.

A gravitational acceleration being independent of a body's mass also follows from the Kepler's third law.

Sure, we need a standard of force, but I don't see how it makes F and m interdependent. I see that F and m are defined separately while F=ma sets their interdependence.

I am not sure it is necessary. We only need to define that in any given conditions (i.e. under given albeit undefined forces) mass is inversely proportional to acceleration. We take a spring, again, stretch it by 10 cm, attach a "unit" of mass, release and measure the acceleration. Then we attach another body to the same spring stretched by the same amount and measure its acceleration. We get the mass of the second body. Etc. Now we have independent definitions of mass and of force and can put them together without any circularity. We could've discovered that \(F \propto m^2a^2\) or \(F \propto e^{ma}\), ... PS. To establish that "mass is inversely proportional to acceleration under given conditions" we can do, e.g., the following. First, the same spring, body, stretch, release, measure acceleration procedure. Then, halve the body and define that its half has half the mass. Repeat the procedure and find that the acceleration doubles. ...