# Why special relativity is unsuitable to describe gravity

I am trying to understand why the special relativity is not suitable for describing the gravity.
Consider a counterexample assuming it is the suitable and the space-time containing a gravitational mass is flat. Then one could describe the acceleration of a test particle from his inertial frame of reference (FOR) where the gravitational mass is located in the center of this coordinates system. If, however, there is another observer moving relative to the first system with a constant speed, he can still calculate the acceleration of the test particle relative to his new coordinates. However, this gives different acceleration because the acceleration is frame dependent according to SR which may not agree with the fact that the different value of new acceleration must also match the value given by ##g=G\frac{M}{r^2}## and ##M## should be larger relative to the moving observer. This drove Einstein to think about the general covariance principle where only the free falling observer has the privilege to consider his frame inertial which leads to considering all other frames non-inertial and hence the curved space-time emerged.
Is this a true argument?

• sweet springs

Ibix
2020 Award
The problem with Newtonian gravity in special relativity is that its effects are instantaneous at any distance, but instantaneous is a frame-dependent thing in relativity. Attempts to fix this by adding a propagation speed to gravity didn't work.

Experimental tests on semi-Newtonian theories also ended up predicting the wrong deflection of light, where GR predicted correctly.

• PeterDonis, Buzz Bloom and vanhees71
Hi. Fake gravitation such as motions of constant acceleration or rotation can be fully understood by applying SR. Real gravity seems to have similar behaviour with fake ones thus introduced GR. Best.

• timmdeeg
FactChecker
Gold Member
I think that another difference is that the SR acceleration vectors would all be parallel whereas gravity "acceleration" vectors are all pointing away from the gravitational mass and are not parallel. And the nature of the vector magnitudes are different in SR and GR. Keeping track of geodesic paths in GR is a significant problem.

PeterDonis
Mentor
2020 Award
this gives different acceleration because the acceleration is frame dependent according to SR which may not agree with the fact that the different value of new acceleration must also match the value given by ##g=G\frac{M}{r^2}## and ##M## should be larger relative to the moving observer.

If you assume that ##M## is relativistic mass, yes, ##M## will be larger in the moving frame. But the coordinate acceleration will also be larger in the moving frame. Also, the distance ##r## will be length contracted in the moving frame. It's not clear just from stating those facts whether things will still end up working out in the moving frame.

the general covariance principle where only the free falling observer has the privilege to consider his frame inertial which leads to considering all other frames non-inertial

That's not the principle of general covariance; in fact it contradicts it, by picking out a special class of frames (the inertial ones). The principle of general covariance says you can pick any frame you like--inertial, non-inertial, it doesn't matter--and the laws of physics will still look the same. Giving a special status to inertial frames is what special relativity does, not general relativity.

• nitsuj
PeterDonis
Mentor
2020 Award
I am trying to understand why the special relativity is not suitable for describing the gravity.

Ibix gave the correct answer: instantaneous propagation speed. That is inconsistent with SR, and trying to fix it by just adding a propagation delay to Newtonian gravity gives a theory that is grossly inconsistent with observations (for example, such a theory predicts that planetary orbits in the solar system should be unstable on fairly short time scales).

Ibix gave the correct answer: instantaneous propagation speed. That is inconsistent with SR, and trying to fix it by just adding a propagation delay to Newtonian gravity gives a theory that is grossly inconsistent with observations (for example, such a theory predicts that planetary orbits in the solar system should be unstable on fairly short time scales).
I dont understand why there should be an instantaneous propagation speed in a solar system where the mass of the sun is stable over the time. I do not feel this is the correct answer, what I missed here?. Even from the history, Einstein looked at it from different view point. He thought that there is no point to restrict ourselves to particular frame of reference when we describe the general motion in physics. And by general motion here, I mean movement with constant velocity or acceleration. This means any arbitrarily frame can describe the law of physics correctly. But the inertial frame in a gravity-free space is also equivalent to a free falling frame according to the principle of equivalence. This give the relation between the general relativity motion and the gravitation. see Stanford encyclopedia section2/5 https://plato.stanford.edu/entries/spacetime-iframes/
This drives me to ask another question, is that all about the gravity alone? how about other general physical movement or actions in physics? is gravity the only force that undermines the concept of general covariance?

Ibix
2020 Award
I dont understand why there should be an instantaneous propagation speed
##F=GMm/r^2## has no propagation term. It implies that a change in the position or mass of one a body should be reflected in its gravitational effect instantly everywhere else in the universe. That's inconsistent with relativity, which doesn't have a universal definition of "instantly".
This drives my to ask another question, is that all about about gravity alone? how about the other general physical movement or actions in physics? is gravity the only force that undermines the concept of general covariance?
The special thing about gravity in the historical sense is that it is the only force for which we ever had a non-relativistic theory. Maxwell's equations are Lorentz-covariant not Galileo-covariant, although that wasn't understood at the time Maxwell developed them, and the strong and weak force theories were developed after Einstein.

So it isn't that gravity "undermines the concept of general covariance", it's that relativity undermines Newtonian gravity. That, in turn, means that Newtonian gravity can only be an approximation to something else that respects relativity's speed limits. That something else turns out to be General Relativity.

• ##F=GMm/r^2## has no propagation term. It implies that a change in the position or mass of one a body should be reflected in its gravitational effect instantly everywhere else in the universe. That's inconsistent with relativity, which doesn't have a universal definition of "instantly".
The special thing about gravity in the historical sense is that it is the only force for which we ever had a non-relativistic theory. Maxwell's equations are Lorentz-covariant not Galileo-covariant, although that wasn't understood at the time Maxwell developed them, and the strong and weak force theories were developed after Einstein.

So it isn't that gravity "undermines the concept of general covariance", it's that relativity undermines Newtonian gravity. That, in turn, means that Newtonian gravity can only be an approximation to something else that respects relativity's speed limits. That something else turns out to be General Relativity.
This mean that not only the change in the sun mass, but also the information about the change in its position still needs to propagate in the form of gravitation wave. If, so can we detect this wave knowing that the sun is moving inside the galaxy?

##F=GMm/r^2## has no propagation term. It implies that a change in the position or mass of one a body should be reflected in its gravitational effect instantly everywhere else in the universe. That's inconsistent with relativity, which doesn't have a universal definition of "instantly".
The special thing about gravity in the historical sense is that it is the only force for which we ever had a non-relativistic theory. Maxwell's equations are Lorentz-covariant not Galileo-covariant, although that wasn't understood at the time Maxwell developed them, and the strong and weak force theories were developed after Einstein.

So it isn't that gravity "undermines the concept of general covariance", it's that relativity undermines Newtonian gravity. That, in turn, means that Newtonian gravity can only be an approximation to something else that respects relativity's speed limits. That something else turns out to be General Relativity.
And more importantly, will the free faller detect the gravitational wave if the sun suddenly disappears? If so, how can he explain that wave?

Nugatory
Mentor
I dont understand why there should be an instantaneous propagation speed in a solar system where the mass of the sun is stable over the time. I do not feel this is the correct answer, what I missed here?
Instaneous propagation is inherent in Newton's ##F=Gm_1m_2/r^2##. To evaluate the force on an object at any moment, you need to know the positions of all the other masses in the universe at that moment - and that assumption implies absolute simultaneity, which is equivalent to instaneous propagation.

Another way of seeing the problem is to consider that force is a vector, so it has a direction. The earth is moving relative to the sun. Does the force vector acting on the earth point along a line between the earth and where the sun is now relative to the earth; or does it point along a line between the earth and where the sun was relative to the earth a moment ago? The former implies instaneous propagation and the latter implies a finite speed of propagation, with larger values of "a moment ago" corresponding to larger propagation delays. Stable orbits (and there is no doubt that the planetary orbits are stable) require that the vector always point along the line between where the sun and the earth are right now with no propagation delay, so even in a system in which the mass of the sun is stable we are stuck with instaneous propagation.

• Ibix
2020 Award
This mean that not only the change in the sun mass, but also the information about the change in its position still needs to propagate in the form of gravitation wave. If, so can we detect this wave knowing that the sun is moving inside the galaxy?
I don't know much about gravitational waves; however I think the answer is that it's not that simple. Just "moving" isn't enough - gravitational waves come from changing quadropole moments, which means that it's motion relative to other masses that's important. I'll have to leave others to answer this.
And more importantly, will the free faller detect the gravitational wave if the sun suddenly disappears? If so, how can he explain that wave?
Energy conservation is built in to Einstein's field equations. They literally cannot describe the Sun just disappearing. If it were to go nova or something, you would notice nothing gravitationally until mass and energy from the nova started passing you, at which point you would have ample explanation for changes in the gravitational field.

Energy conservation is built in to Einstein's field equations. They literally cannot describe the Sun just disappearing. If it were to go nova or something, you would notice nothing gravitationally until mass and energy from the nova started passing you, at which point you would have ample explanation for changes in the gravitational field.
So the free faller will not notice the gravitational wave. Why? the wave is a physical wave propagates outward from the center of the sun and the faller moves inward to the center of the sun, so logically they should meet!

Ibix
2020 Award
So the free faller will not notice the gravitational wave.
I did not say that. I just said that there's no change in gravity at your location until some of the energy and matter from a stellar explosion has passed you. Gravitational waves and electromagnetic radiation travel at the same speed - c, in a vacuum.

Instaneous propagation is inherent in Newton's ##F=Gm_1m_2/r^2##. To evaluate the force on an object at any moment, you need to know the positions of all the other masses in the universe at that moment - and that assumption implies absolute simultaneity, which is equivalent to instaneous propagation.

Another way of seeing the problem is to consider that force is a vector, so it has a direction. The earth is moving relative to the sun. Does the force vector acting on the earth point along a line between the earth and where the sun is now relative to the earth; or does it point along a line between the earth and where the sun was relative to the earth a moment ago? The former implies instaneous propagation and the latter implies a finite speed of propagation, with larger values of "a moment ago" corresponding to larger propagation delays. Stable orbits (and there is no doubt that the planetary orbits are stable) require that the vector always point along the line between where the sun and the earth are right now with no propagation delay, so even in a system in which the mass of the sun is stable we are stuck with instaneous propagation.
I got it, thank you. But again, why the absolute simultaneity does not hold in general relativity where the sun bends the space around it far away from the center?

Nugatory
Mentor
I got it, thank you. But again, why the absolute simultaneity does not hold in general relativity where the sun bends the space around it far away from the center?
GR describes the motion of a body based on the spacetime curvature at the point where the body is at a given moment; the positions of distant objects at that moment don't matter. Of course that spacetime curvature is affected by where the distant objects were in the past, but that is consistent with a propagation delay so we don't need absolute simultaneity.

You might reasonably ask how we can have stable orbits with propagation delay in GR but we can't have stable orbits with propagation delay with Newtonian gravity. The answer is that GR predicts different (only very slightly different for planets orbiting a sun-sized mass) orbits than Newtonian gravity; we see this in the precession of Mercury's orbit because it is closest to the sun and the tiny difference is most visible there. The GR-predicted orbits are stable even with a propagation delay.

Buzz Bloom
Gold Member
The GR-predicted orbits are stable even with a propagation delay.
Hi Nugatory:

I am guessing that you are intentionally for pedagogical reasons ignoring the fact that gravitational waves (GWs) make orbits unstable in the sense that over time an orbit shrinks due to the loss of kinetic energy from the moving bodies due to the GWs they generate.

Regards,
Buzz

Dale
Mentor
2020 Award
I dont understand why there should be an instantaneous propagation speed in a solar system where the mass of the sun is stable over the time.
There are other gravitating objects in the solar system besides the sun.

FactChecker
Gold Member
Is it correct to think that in a Sun-centered coordinate system, the time-space distortion is essentially fixed and not "propagating". (Ignoring smaller gravitational objects.) Alternatively, in a planet-centered coordinate system, the Sun is moving and the time-space distortion effects propagate at the speed of light?

Buzz Bloom
Gold Member
I don`t understand why there should be an instantaneous propagation speed in a solar system where the mass of the sun is stable over the time.

Another small nit. The mass of the sun is not stable over time. The sun is constantly losing mass, and it has been doing so almost ever since it came into existence. The photon energy it radiates, as well as the mass particles it radiates, reduce it's mass continuously. (There may have been a relatively short period when the mass of in falling matter exceeded the mass loss.)

Regards,
Buzz

Also one more conflicting point for me. If the sun is also considered to be a free faller relative to the centre of galaxy, then the coordinates of it should be linear and the space-time is flat but we know that the coordinate of the solar system is curvilinear and the space-time is curved. So how the solar system has two types of space-time at the same moment?

Nugatory
Mentor
If the sun is also considered to be a free faller relative to the centre of galaxy, then the coordinates of it should be linear and the space-time is flat but we know that the coordinate of the solar system is curvilinear and the space-time is curved. So how the solar system has two types of space-time at the same moment?
It doesn't. Free-fall doesn't mean flat spacetime, it means that the free-falling object is experiencing no forces (electromagnetic, rocket thrust, earth supporting it from underneath, rope supporting it from an overhead beam....) to pull it off its natural path through spacetime.

It doesn't. Free-fall doesn't mean flat spacetime, it means that the free-falling object is experiencing no forces (electromagnetic, rocket thrust, earth supporting it from underneath, rope supporting it from an overhead beam....) to pull it off its natural path through spacetime.
What I learnt is that the inertial space is locally flat like a flat tangent surface to the curved surface of a sphere for example. So if the bigger sphere is the galaxy, the free falling Sun is then a locally flat space. But again the sun is another curved spacetime surface. Does it mean there is no universal flat space because each free falling observer can consider his frame locally flat? In other words, do we need to know all of information about the distribution of masses in the universe before speaking about the flatness of the space?

Buzz Bloom
Gold Member
Free-fall doesn't mean flat spacetime, it means that the free-falling object is experiencing no forces (electromagnetic, rocket thrust, earth supporting it from underneath, rope supporting it from an overhead beam....) to pull it off its natural path through spacetime.
Hi Nugatory:
I think there is a vocabulary usage here that confuses me. As I understand the answers to the questions I asked in another thread
depending on the usage, "the natural path" may or may not include the effects of Gravitational Waves (GWs).

If a body has negligible mass, then the deviation from "the natural path" is negligible, and is frequently ignored. However, is the body has any mass, there is always some deviation in the average distance from another massive body. In a binary system, the "natural" GR elliptical orbit (with precession of the periapsis) shrinks due to energy loss from GWs.

Regards,
Buzz

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Nugatory
Mentor
Does it mean there is no universal flat space because each free falling observer can consider his frame locally flat?
There is no universal flat space because the whole thing is curved (and all observers everywhere agree about how much it is curved at each point). However, any sufficiently small region can be approximated as if it were flat. If your measuring instruments are good enough to detect curvature across a distance of 1000 kilometers, try them on a distance of 1 kilometer, and if they still detect the curvature you can try them across a distance of one meter - and eventually you'll find just how local a region has to be before it can be considered locally flat. There's an analogy with the curved surface of the earth here: we know it is curved everywhere, but if you and I are laying out the foundations of houses on small plots of land on opposite sides of the earth, we'll both pretend that the earth is flat and ignore the fact that the surfaces of the two plots are not parallel so do not share a common flat space.
In other words, do we need to know all of information about the distribution of masses in the universe before speaking about the flatness of the space?
No, because we can ignore the contributions from things that there are far away and happening outside of our past light cone. Practically speaking, the Andromeda galaxy's contribution to curvature in the vicinity of the earth is altogether negligible compared with the contributions from the other planets in the solar system, the sun, the moon, and the earth's own mass. However, if we really do include everything, there will always be some tiny curvature that in principle could be detected with sufficiently sensitive instruments.

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