General Relativity vs Newtonian Mechanics

[ueit]

I am involved in a discussion about non-locality in QM. I'm arguing that it is possible for a local mechanism to emulate a non-local one and the best example I could find is gravity (the apparently non-local force required by Newtonian gravity being explained by a local mechanism in GR).

My question is if Newtonian gravity (where the force points toward the instantaneous position of each massive body) is a good approximation for all types of motion, including non-uniform accelerations. To give an example, suppose that a planet as big as Mars, coming from outside the Solar system, passes near Mars so that Mars' orbit is significantly changed. In this case, would Earth be accelerated immediately towards the new position of Mars or it will continue to accelerate towards Mars' retarded position until the signal, coming at the speed of light would arrive?

Thanks!

 

[Gib Z]

It would continue until the signal arrived, how ever the effect of this delay and therefore accuracy of the Mechanics depends on the time the signal takes, ie distance between the planets.

 

[pervect]

The short answer is no.

Try reading

http://math.ucr.edu/home/baez/physics/Relativity/GR/grav_speed.html

I'll quote some of the relevant sections.

Strictly speaking, gravity is not a "force" in general relativity, and a description in terms of speed and direction can be tricky. For weak fields, though, one can describe the theory in a sort of Newtonian language. In that case, one finds that the "force" in GR is not quite central--it does not point directly towards the source of the gravitational field--and that it depends on velocity as well as position. The net result is that the effect of propagation delay is almost exactly cancelled, and general relativity very nearly reproduces the Newtonian result.

This cancellation may seem less strange if one notes that a similar effect occurs in electromagnetism. If a charged particle is moving at a constant velocity, it exerts a force that points toward its present position, not its retarded position, even though electromagnetic interactions certainly move at the speed of light. Here, as in general relativity, subtleties in the nature of the interaction "conspire" to disguise the effect of propagation delay. It should be emphasized that in both electromagnetism and general relativity, this effect is not put in ad hoc but comes out of the equations. Also, the cancellation is nearly exact only for constant velocities. If a charged particle or a gravitating mass suddenly accelerates, the change in the electric or gravitational field propagates outward at the speed of light.

Since this point can be confusing, it's worth exploring a little further, in a slightly more technical manner...

 

[ueit]

The short answer is no.

Try reading

http://math.ucr.edu/home/baez/physics/Relativity/GR/grav_speed.html

I'll quote some of the relevant sections.

So, I understand that your answer is no, Newtonian gravity is not a good approximation for all types of motion. But, this seems to contradict the fact that Newtonian gravity can be succesfully used to describe non-circular orbits or multi-body systems like Sun-Earth-Moon in spite of non-uniform accelerations being present. Is the motion, in these situations, close enough to constant velocity?

Another question is if the gravitational radiation gives us a quantitative estimation of the error between the trajectory computed using GR and Newtonian theory.

 

[JesseM]

So, I understand that your answer is no, Newtonian gravity is not a good approximation for all types of motion. But, this seems to contradict the fact that Newtonian gravity can be succesfully used to describe non-circular orbits or multi-body systems like Sun-Earth-Moon in spite of non-uniform accelerations being present. Is the motion, in these situations, close enough to constant velocity?

Another question is if the gravitational radiation gives us a quantitative estimation of the error between the trajectory computed using GR and Newtonian theory.

I don't think the fact that GR does not predict attaction towards the current position necessarily means that Newtonian gravity is "not a good approximation". After all, suppose you designed a modified version of Newtonian gravity where the gravitational force on the Earth from another planet pointed, not at a planet's current position, but at the position it would be if you "extrapolated" its motion from the last moment it was in the past light cone of the Earth at the present time, with the extrapolation based either on the planet's instantaneous velocity or on its instantaneous acceleration at the last moment it was still in the light cone (I think the latter would be closer to the type of 'extrapolation' which Baez describes GR doing, but I'm not sure). Considering how small the time it takes for light to travel from any other planet in the solar system to Earth when compared with the time for that planet to complete an entire orbit, I would think that the differences between the predictions of a modified Newtonian theory like this and actual Newtonian theory would be quite small, and thus that instantaneous Newtonian gravity would be quite a good approximation for such a modified theory.

Obviously GR is not the same as the type of modified Newtonian theory I describe, but if I'm understanding Baez correctly it should at least be similar in the way it "extrapolates" the direction that distant objects pull on you.

Speaking of which, a question came up on the other thread about what types of motion cause gravitational waves and which don't--Baez says that any constant acceleration will emit no waves:

Similarly, in general relativity, a mass moving at a constant acceleration does not radiate (the lowest order radiation is quadrupole), so for consistency, an even more complete cancellation of the effect of retardation must occur. This is exactly what one finds when one solves the equations of motion in general relativity.

Whereas the wikipedia page seems to say that only spherically symmetric or cylindrically symmetric acceleration avoids emitting waves:

In general terms, gravitational waves are radiated by objects whose motion involves acceleration, provided that the motion is not perfectly spherically symmetric (like a spinning, expanding or contracting sphere) or cylindrically symmetric (like a spinning disk).

Obviously I'd tend to trust Baez over wikipedia, but I wonder if I'm not just misunderstanding here--are they saying the same thing? For example, perhaps it's impossible to have "uniform acceleration" in GR that is not also spherically/cylindrically symmetric, like a planet in a circular orbit? Can anyone clarify?

 

[pervect]

So, I understand that your answer is no, Newtonian gravity is not a good approximation for all types of motion. But, this seems to contradict the fact that Newtonian gravity can be succesfully used to describe non-circular orbits or multi-body systems like Sun-Earth-Moon in spite of non-uniform accelerations being present. Is the motion, in these situations, close enough to constant velocity?

Another question is if the gravitational radiation gives us a quantitative estimation of the error between the trajectory computed using GR and Newtonian theory.

In the case of a circular or near circular orbit, gravitational radiation will give us a quantitative estimate of how long it takes a body to inspiral.

The Hulse-Taylor measurements of a binary pulsar won a nobel prize for confirming GR's prediction of the rate of inspiral.

Think about it in Newtonian terms. As the FAQ mentions, if you have a central force, angular momentum is conserved. If you do not have a central force, you have a change in angular momentum. This is either an inspiral or an outspiral. GR predicts an inspiral - energy is lost to gravitational waves.

As far as accuracy goes, we are perfectly justified in neglecting the effects of gravitational radiation in solar system experiments.

A very crude estimate of the characteristic time it takes a system to lose its energy by gravitational radiation is

$$t_{react} = \left( \frac{2 R} {R_s} \right)^\frac{5}{2} T_{orbit}$$

where $t_{orbit}$ is the orbital period, and $R_s$ is the Schwarzschild radius of the central mass. See MTW, "Gravitation", pg 981 for the origin of this estimate - note that I've taken the liberty to convert the equation from geometric units into regualr units.

You can model the energy lose in a given short time interval as

energy loss = current energy * (short time interval) / $t_{react}$If we look at the case of the Earth, R_s for the sun is 3 km. So R/R_s is 5*10^7, and we get a characteristic time $t_{react}$ of 10^20 years. So GR predicts an unmeasureably small effect for the Earth.

You can redo this for Mercury if you like, but you still won't get anything measurable.

Note that of course the Hulse-Taylor binary pulsar measurements show that under extreme enough conditions, the effect has been observed to be measurable. We just don't encounter such conditions in the solar system.

Gravitational radiation will not explain some other predicted GR effects that affect trajectories, however, such as the doubling of the "Newtonian" value for the deflection of light, or the precession of the perihelion of Mercury.

Thus it would be incorrect to think of gravitational radiation as the sole difference between GR and Newtonian theory as far as trajectories go, though it is the only effect that affects inspiral times.