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When I was in college, I posed to my astronomy professor a thought experiment that had been puzzling my mind for some time: "If all the matter in the Sun magically disappeared instantly, how long would it take its gravity to stop having an effect on us?" His response was that the force of gravity is instant, unlike the speed of light, which appears instant.
My big question is: "Do we know it's instant?" We can't possibly move an object large enough to have a noticeable gravitational influence fast enough to measure if it creates (or doesn't create) a doppler-like phenomenon.
$\begingroup$From what I know, it's not instant. Every piece of information, including that due to gravity, travels at most at the speed of light. See this for example: en.wikipedia.org/wiki/Action_at_a_distance#Gravity$\endgroup$
$\begingroup$No, it travels at the speed of light. If the sun magically disappeared it would take 9 minutes until the Earth stopped feeling the gravity.$\endgroup$
The first question as stated has a rather trivial answer:
"If the sun magically disappeared, instantly, along with all its influences, how long would it take its gravity to stop having an effect on us?"
Since the Sun's gravity is among its influences, it would instantly stop having an effect on us. That's just part of the magical situation, and doesn't even involve any physics. A bit more interesting is the question without the bolded part.
In general relativity, changes in the gravitational field propagate at the speed of light. Thus, one might expect that the magical and instant disappearance of the Sun would not affect earth for about eight minutes, since that's how long light from the Sun takes to reach Earth.
However, this is mistaken because the instant disappearance of the Sun itself violates general relativity, as the Einstein field equation enforces a kind of local conservation law on the stress-energy tensor analogous to the non-divergence of the magnetic field in the electromagnetism: in any small neighborhood of spacetime, there are no local sources or sinks of stress-energy; it must come from somewhere and go somewhere. Since the magical instant disappearance of the Sun violates general relativity, it does not make sense to use that theory to predict what happens in such a situation.
Thus, the Sun's gravity instantly ceasing any effect on the Earth is just as consistent with general relativity as having any sort of time-delay. Or to be precise, it's no more inconsistent.
My big question, now, is: "How do we know it's instant?"
It's not instant, but it can appear that way.
We can't possibly move an object large enough to have a noticeable gravitational influence fast enough to measure if it creates (or doesn't create) a doppler-like phenomenon.
We don't have to: solar system dynamics are quite fast enough. An simple calculation due to Laplace in the early nineteenth century concluded that if gravity aberrated, Earth's orbit would crash into the Sun on the time-scale of about four centuries. Thus gravity does not aberrate appreciably--more careful analyses concluded that in the Newtonian framework, the speed of gravity must be more than $2\times10^{10}$ the speed of light to be consistent with the observed lack of aberration.
This may seem quite a bit puzzling with how it fits with general relativity's claim that changes in the gravitational field propagate at the speed of light, but it's actually not that peculiar. As an analogy, the electric field of a uniformly moving electric charge is directed toward the instantaneous position of the charge--not where the charge used to be, as one might expect from a speed of light delay. This doesn't mean that electromagnetism propagates instantaneously--if you wiggle the charge, that information will be limited by $c$, as the electromagnetic field changes in response to your action. Instead, it's just something that's true for uniformly moving charges: the electric field "anticipates" where the change will be if no influence acts on it. If the charge velocity changes slowly enough, it will look like electromagnetism is instantaneous, even though it really isn't.
Gravity does this even better: the gravitational field of a uniformly accelerating mass is toward its current position. Thus, gravity "anticipates" where the mass will be based on not just current velocity, but also acceleration. Thus, if conditions are such that the acceleration of gravitating bodies changes slowly (as is the case in the solar system), gravity will look instantaneous. But this is only approximately true if the acceleration changes slowly--it's just a very good approximation under the conditions of the solar system. After all, Newtonian gravity works well.
A detailed analysis of this can be found in Steve Carlip's Aberration and the Speed of Gravity, Phys.Lett.A 267:81-87 (2000) [arXiV:gr-qc/9909087].
If he was wrong, how do we know it's not?
We have a lot of evidence for general relativity, but the best current evidence that gravitational radiation behaves as GTR says it does is Hulse-Taylor binary. However, there is no direct observation of gravitational radiation yet. The connection between the degree of apparent cancellation of velocity-dependent effects in both electromagnetism and gravity, including its connection with the dipole nature of EM radiation and quadrupole nature of gravitational radiation, can also be found in Carlip's paper.
$\begingroup$sorry, by "along with all its influences", I meant that there wouldn't be, like, a gravity well or anything left where it was. I didn't mean its influences would be magically removed from the entire universe.$\endgroup$
$\begingroup$If I understand this correctly, we appear to experience a gravitational force directed at the point where the Sun is now, not where it was 8 minutes ago (the latter is where we see it). But if the Sun's core suddenly collapsed, and we were able to detect the resulting gravitational waves, we'd detect them 8 minutes after the event, and from a direction matching the visible position of the Sun, not from the direction of the apparent gravitational force. Is that correct?$\endgroup$
$\begingroup$@KeithThompson yes, that's right, with minor caveats. We'd detect gravitational waves if the Sun's quadrupole moment changes violently in the collapse about 8 minutes afterward. (I phrase things this way because, e.g., a spherically symmetric core collapse wouldn't gravitationally radiate.)$\endgroup$
$\begingroup$@Tanenthor I know "rapidly changing" is not exactly accurate, but it is the best notice I can put on this post to indicate that it hasn't taken account of latest results.$\endgroup$
Updated 2018: @Stan Liou's answer is excellent, but since he wrote that answer, we have accurately measured the speed of gravitational waves and confirmed to a very high degree of accuracy that it is the speed of light in vacuum.
In August 2017, LIGO observed GW170817, a neutron star inspiral. X-ray observatories in orbit detected x-rays from the collision with at most a 2-second difference in travel time, so speed-of-light == speed-of-gravitation is confirmed. Here's a decent summary of the results.
$\begingroup$The question was about static gravity not gravitational radiation. The latter is only created if the quadrupole moment of the object changes. Letting a point- or spherical mass just disappear only changes the monopole moment, so this should not produce gravitational radiation$\endgroup$
$\begingroup$@Thomas There's no real distinction between static gravity and gravitational radiation: it's all curvature of spacetime. The speed of gravitational radiation (the speed of light) is the speed at which changes to a gravitational field -- such as that due to the Sun disappearing -- propagate, If the Sun disappeared it would take about eight minutes before the change to spacetime rippled out as far as the Earth. Key point: Earth is not held in orbit by a force from the Sun. Earth falls freely through the space-time out here when we are.$\endgroup$
Actually, "instant" gravity was part of Newton's theory of gravity. It is now understood that the "speed of gravity" is equal to the speed of light according to general relativity. http://en.wikipedia.org/wiki/Speed_of_gravity
$\begingroup$Cool! So the Earth would, in my scenario, continue orbiting around nothing for ~8 minutes before being flung into who knows where?$\endgroup$
$\begingroup$Yes, I believe that is what would happen. However, we would not notice that the sun was missing until the ~8 minute mark because the light would not have made it to us either. If the sun were to go away, we would not notice (light or gravity) until about 8 minutes later (if general relativity is right).$\endgroup$
$\begingroup$There's still a little room to quibble: "Their actual figure was 1.06 times the speed of light, but there was an error of plus or minus 0.21."$\endgroup$
None of us would know that the Sun had actually disappeared until the approximate eight minute point had passed. At that point, we would all be very quickly screaming and panicking at the absolute and total darkness which would suddenly replace all normal daylight. The gravitational concerns would be the least of our worries, I think.
If it happened at night, those of us in artificial light would be spared the surprise briefly. Those outside might be wondering why the Moon suddenly winked out... as eyes became used to the total darkness, stars in clear areas of the sky would be all that is visible, anywhere.
The Earth would freeze, PDQ; we would not last long.
$\begingroup$I didn't ask Randall Munroe; I don't want a "what if the sun disappeared" answer, but a "how do we know gravity is instant" answer, which has been provided. -1 for not adding anything to the conversation with your answer.$\endgroup$
$\begingroup$@Supuhstar you could be a bit more helpful and instructive to help a new user make better answers instead of being dismissive. I note you are fairly new yourself. As a counterpoint, your question was not a 'good' question, in that it showed little prior research, and asked about a personal understanding that wasn't correct. Please check the FAQS for how to ask good questions.$\endgroup$
Gravitational waves have been detected over the past couple of years, and these are simply ripples in gravitational fields that propagate at the speed of light. We know they propagate at the speed of light because that's how gravity wave detectors like LIGO work, and because we have correlated some of the gravity signals to light signals that arrive at the same time.
