What is the fastest possible acceleration?

Context & Question

In physics, the fastest possible speed is accepted as the speed of light. No other mass has been recorded to travel as fast as the photon. What is the acceleration of the photon? How long does it take for it to get to its top speed, from emission from its source to its constant velocity through the medium of space-time? Does the photon also hold the record for fastest acceleration? Taking these questions into account, I would like to probe further and ask if instantaneous acceleration is indeed possible by the laws of physics?

• It doesn't get to top speed. It is top speed all the time. – DanielC Oct 26 '18 at 20:31
• Possible duplicates: physics.stackexchange.com/q/20289/2451 , physics.stackexchange.com/q/94049/2451 and links therein. – Qmechanic Oct 26 '18 at 21:44
• This is a fun question, because it forces one to think carefully about what is meant by "acceleration". We think of refractive index as inversely related to the speed of light in a medium. It's not entirely unreasonable to consider that a light wave "accelerates" as it moves from a high-index medium to a lower-index medium. – S. McGrew Oct 26 '18 at 22:07
• by instantaneous acceleration you mean infinite acceleration, right? because instantaneous acceleration exists, it is the derivative with respect to time of the instantaneous speed – Wolphram jonny Oct 26 '18 at 22:14
• possible duplicate physics.stackexchange.com/q/3334 – Wolphram jonny Oct 26 '18 at 22:15

The smallest amount of time is the Planck time which is approximately $$10^{-43}$$ seconds. The speed of light is approximately $$10^8$$ meters per second and so the fastest acceleration is approximately $$10^8/10^{-43}=10^{51}$$ meters per second$$^2$$.

• Having strong doubts. The Planck units are in my understanding regimes where all known theories reach their final point of break-down. Moreover, acceleration = speed / time is a formula whose validity I doubt - when so strong forces apply there are general relativistic effects we need to take into consideration. – Nobody-Knows-I-am-a-Dog Nov 9 '18 at 23:44
• Its a back of the envelope estimate but (acceleration = speed / time) still holds in special relativity provided the velocity and time values are in the same frame. – Virgo Nov 10 '18 at 2:51
• Your estimate fails due to GENERAL relativistic effects. So high an acceleration means so much localized energy and thus mass and thus gravity that your time scale will be heavily affected; the length distortions will depend on the direction of movement and further complicate matters. I did not carry out the calculations but the region might even collapse into a black hole. Thus, I think, this "back of the envelope estimate" is pretty meaningless and does not provide a physical answer to the OP. Mathematically one can do all kinds of extrapolations outside physically meaningful ranges... – Nobody-Knows-I-am-a-Dog Nov 10 '18 at 19:34
• That's the whole point of using the Planck time. My estimate will be when a black hole forms, so it gives an upper bound to the possible acceleration. It is only going to be order of magnitude which I think is what the questioner was after. – Virgo Nov 11 '18 at 4:42

If a photon can be said to change speed at a boundary between two optical materials (e.g., glass and air), then you might ask how quickly it changes speed at that boundary. However, no physical boundary is infinitesimally thin, so at least on a sub-femtosecond timescale, the effective speed of a photon changes somewhat gradually while crossing a physical boundary. But as the other answers have indicated, the "speed of a photon" is not precisely definable in a physically useful way because a photon is not really a point particle; it's a spread-out wavepacket. If the speed can't be defined precisely, the acceleration can's be defined precisely.

You didn't ask about acceleration of a point charge like an electron. We can come a lot closer to defining velocity and acceleration for an electron, but ultimately run into the same limitations due to the fact that an electron, too, is really a wavepacket. However in the case of an electron there is a further limitation: an accelerating charge radiates EM radiation at a rate proportional to the square of its acceleration See Larmor Radiation. So, the quicker you try to change its speed, the more it will cost you in expended energy. It would take an infinite amount of power to provide infinite acceleration to even a massless point charge if there were such a thing.

The speed of light (which doesn't have quite the same meaning as the speed of a photon) is a property of the vacuum of space, "inhabited" by electromagnetic fields described by Maxwell's equations. An electromagnetic wave (not quite the same thing as a photon) moves at a speed determined by the properties of the matter in space the wave is moving through, so "acceleration" of an EM wave (i.e., change of speed or direction) can always be traced directly to a change in material properties along the spread-out "path" of the wave.

• As you state that a given EM wave's "acceleration" is dependent on its medium (our universe's "space-time" as described by Albert Einstein being the medium) I would like to pose a further inquiry into the behavioral limitations of matter in the theorized "11-dimensional space" between universes. This refers to the aspect of Inflation theory in which a universe "bubble" buds off of and into other "parallel" universes To our best knowledge how would an EM wave behave in this 11-dimensional space? I don't ask about other universes as as far as I know we do not have respective dimensional theories – MJMarquez Oct 27 '18 at 1:30
• I'd suggest posing that as a new and separate question. – S. McGrew Oct 27 '18 at 3:25

Your questions is tricky (and interesting) since it is located on the boundary of established theories.

Following the assumptions of relativity theory: The photon never exists in a state where it has a different speed than the (local) speed of light for all reference frames. It "is born" like that. Since the photon is measured at the same speed from every frame our usual concepts of acceleration and speed break down and it is doubtful if extrapolation from below-light-speed makes sense.

Following the assumptions of quantum electro dynamics (QED): The photon does not take a single path which allows us to determine a speed or an acceleration. It takes (at least from the perspective of the employed mathematical procedure which comes up with the correct probabilities) all possible paths from emitter to detector at the same time, we must sum up all the probability amplitudes along all these paths and we get a probability at which detector the photon pops up into existence. Statements about paths taken make no sense since we cannot measure them in principle.

Probably there is such a thing like maximal acceleration. Accelerating an object for a certain distance requires energy in a small location near that object. Too much energy means too much mass. Too much localized mass produces a black hole. So there probably is a limit. Still: We are debating a situation where we are widely outside of the range in which our existing theories had been tested. So a break down of our physical concepts is quite likely as well.