Gravitational force equation derived from GR

[sha1000]

TL;DR Summary

Gravitational force in GR

Hello everyone,

I know that GR equations are complicated and beyond my scope.

But does GR give a simple gravitational equation: Force (as we know it) as a function of distance? (without any complicated tensors).

- If yes. What is the equation? Does it give us something similar to Newtons equation?

- If no. Can we make some sort of relativistic corrections to Newtons equation and make it work?

Thank you in advance.

 

[PeterDonis]

does GR give a simple gravitational equation: Force (as we know it) as a function of distance?

Not in general, no, but that's because in the general case, gravity is not a Newtonian force and the Newtonian picture of gravity as an inverse square force is physically wrong, so you wouldn't expect GR to give what you are asking for.

GR does reduce to Newtonian gravity in the appropriate special case, which is a spherically symmetric gravitating mass in the weak field, slow speed approximation. "Weak field" means, roughly speaking, that the gravitational potential energy is much, much smaller than the rest mass energy of all gravitating bodies, and "slow speed" means all relative speeds are much smaller than the speed of light. In that approximation, GR just says that gravity looks like Newtonian gravity.

 

[Orodruin]

It is worth noting that the equations of motion for a test particle in the Schwarzschild spacetime do end up being very similar to the Newtonian case with the addition of an angular momentum dependent potential proportional to $1/r^3$. There are of course some caveats in this, such as the time involved in the equation of motion being the proper time of the test particle and not the global time coordinate and the $r$ coordinate being defined by the area of a spherical shell and not being the physical radius.

 

[Ibix]

If no. Can we make some sort of relativistic corrections to Newtons equation and make it work?

There is a model called the Parameterised Post-Newtonian (PPN) model that produces more precise estimates than Newton. It's still a weak field approximation (so don't try it with black holes), but I believe it will allow you to factor in things like frame dragging.

I think it's a useful calculation tool for weak fields, but if I understand its range of validity correctly I don't think it's all that interesting unless you're into super detailed modelling of orbits around regular stars and planets. It certainly won't help with more exotic scenarios like black holes or cosmology.

 

[PeterDonis]

It's still a weak field approximation

And slow speed, and for the case of an isolated system. So still limited in application compared to GR in general.

 

[sha1000]

Ok its more clear now.

Thank you all for your kind responses.

 

[Mister T]

Can we make some sort of relativistic corrections to Newtons equation and make it work?

In general, the notion of "relativistic corrections" can present a misconception. We don't take Newtonian physics and make corrections to it. Instead, we see that Newtonian physics is just an approximation to relativistic physics that applies under certain conditions. It is that Newtonian approximation that we study when we are studying Newtonian physics.

 

[A.T.]

Summary:: Gravitational force in GR

Hello everyone,

I know that GR equations are complicated and beyond my scope.

But does GR give a simple gravitational equation: Force (as we know it) as a function of distance? (without any complicated tensors).

- If yes. What is the equation? Does it give us something similar to Newtons equation?

- If no. Can we make some sort of relativistic corrections to Newtons equation and make it work?

Thank you in advance.

Here is an old thread about the proper acceleration of a hovering object in GR:

https://www.physicsforums.com/threa...on-in-general-relativity.402135/#post-2711365

The force needed to hover it would be the proper acceleration times its mass.

 

[sha1000]

Here is an old thread about the proper acceleration of a hovering object in GR:

https://www.physicsforums.com/threa...on-in-general-relativity.402135/#post-2711365

The force needed to hover it would be the proper acceleration times its mass.

So basically there is an equation of the proper acceleration of a hovering object which can be derived from GR?

GM/R^2*sqrt(1 - 2GM/R) with c = 1

Is it correct? This can be derived without weak field approximation?

 

[Ibix]

Is it correct?

Assuming that you are hovering above an object that is far from any other object and isn't rotating or carrying any charge, yes. In any other circumstance it's an approximation. And note that GR is not a linear theory like Newtonian gravity, so you can't analyse a scenario with two masses by simply taking that formula, applying it for two $R$ and $M$ values, and adding the results as you could in Newtonian gravity.

 

[Ibix]

And also assuming that you mean $\frac{GM}{r^2\sqrt{1-2GM/r}}$, which is not necessarily what your formula means. Do use LaTeX. It removes ambiguity.

 

[DrGreg]

Assuming that you are hovering above an object that is far from any other object and isn't rotating or carrying any charge, yes. In any other circumstance it's an approximation.

And also assuming that you mean $\frac{GM}{r^2\sqrt{1-2GM/r}}$,

This also assumes that the mass of the hovering object is small enough that we can ignore any gravitation generated by itself.

And $r$ isn't exactly the distance between the two objects. If the large object is a black hole, the distance isn't even defined. If it's a material object such as a star or planet, it will usually be a good approximation to the exact distance. $r$ is exactly the circumference of a circle around the large object passing through the small object, divided by $2\pi$.

 

[vanhees71]

In GR the equation of motion of a "test particle" is given by the geodesic equation in the spacetime determined by the "gravitating" bodies around it (e.g., a planet moving around the Sun). That means that when considered the motion of the test particle in a local inertial frame that no force acts on it.

That's making the "gravitational force" from the point of view of GR like the inertial forces in an accelerated reference frame in Newtonian physics, and that's precisely the way Einstein came to GR, i.e., it makes the gravitational force on a test particle an inertial force, and in a local (!) inertial reference frame (i.e., a frame falling freely with the test particle) there acts no force on the test particle and thus it's going the "straightest line", i.e., a geodesic in the general relativistic spacetime, whose (pseudo-Riemannian) geometry is determined by the gravitating bodies. That's the content of the weak equivalence principle as used to derive GR providing this geometrical picture of the gravitational interaction.

 

[pervect]

I should mention that aside from the Post-Newtonian approximation that has been mentioned, there's a less accurate, but arguably simpler approximation, called the Newtonian approximation. One sees occasional youtube videos based on the idea of using this approximation to "explain" GR. Like other, more sophisticated approximations, it omits some of the more interesting effects of GR, and is not good for strong fields like black holes, though typical formulations of the Newtonain approximation do include some notion of "gravitational time dilation", one of many of the effects of GR that are not really able to be modeled by a force.

 

[sha1000]

Assuming that you are hovering above an object that is far from any other object and isn't rotating or carrying any charge, yes. In any other circumstance it's an approximation. And note that GR is not a linear theory like Newtonian gravity, so you can't analyse a scenario with two masses by simply taking that formula, applying it for two $R$ and $M$ values, and adding the results as you could in Newtonian gravity.

1) And why there is a need to precise these assumptions and approximations for this particular GR equation of the proper acceleration?
I mean, these assumptions must be made even for the classical Newtonian equation.

2) So there is no need to make the weak field assumption?

3) Can we derive the Force equation (as we know it) from the proper acceleration?

 

[sha1000]

This also assumes that the mass of the hovering object is small enough that we can ignore any gravitation generated by itself.

And $r$ isn't exactly the distance between the two objects. If the large object is a black hole, the distance isn't even defined. If it's a material object such as a star or planet, it will usually be a good approximation to the exact distance. $r$ is exactly the circumference of a circle around the large object passing through the small object, divided by $2\pi$.

I have difficulty to understand why we can't give a simple definition of the distance R in GR.

We can measure the real distance between the center of the distant galaxy and any star in the system. Am I wrong?

 

[PeterDonis]

I have difficulty to understand why we can't give a simple definition of the distance R in GR.

You can give one for ordinary systems (i.e., ones not containing a black hole). It's just not the same simple definition as the one in Newtonian mechanics. Newtonian mechanics assumes that the geometry of space is Euclidean. But it's actually not in the presence of gravity. GR includes the non-Euclideanness of space in its model, and that has to be taken into account when defining distances.

 

[PeterDonis]

We can measure the real distance between the center of the distant galaxy and any star in the system. Am I wrong?

How would you measure it? You don't have a ruler that's thousands of light years long that you can place at rest in the galaxy and read off a distance. (And that's leaving out the fact that there are black holes at the centers of galaxies, and there is no meaningful concept of "distance from the center" for a black hole.)

In practice, we estimate distances to distant objects and between distant objects using indirect methods. We don't directly measure any distances of astronomical objects except for those that are close enough to get a measurable parallax.

 

[PeterDonis]

these assumptions must be made even for the classical Newtonian equation.

I'm not sure what assumptions you're talking about, but Newtonian gravity does not have any analogues of the weak field or slow motion assumptions that you need to make in GR (more precisely, in the Schwarzschild solution in GR) to get the Newtonian approximation.

2) So there is no need to make the weak field assumption?

There is if you want to obtain the Newtonian equations as an approximation.

3) Can we derive the Force equation (as we know it) from the proper acceleration?

The proper acceleration being referred to is not due to gravity. It's due to whatever is pushing on the object so it "hovers" at a constant altitude instead of freely falling, and whatever that is will be a non-gravitational force. Objects moving solely under the influence of gravity have zero proper acceleration.

 

[PeterDonis]

GM/R^2*sqrt(1 - 2GM/R) with c = 1

Is it correct? This can be derived without weak field approximation?

This particular equation does not require the weak field approximation, no. It is correct for any object hovering motionless at radial coordinate $R$ outside the horizon of a non-rotating black hole. But, as I noted in my previous post, the proper acceleration of this object is not due to gravity; objects moving solely under gravity have zero proper acceleration.

 

[sha1000]

This particular equation does not require the weak field approximation, no. It is correct for any object hovering motionless at radial coordinate $R$ outside the horizon of a non-rotating black hole. But, as I noted in my previous post, the proper acceleration of this object is not due to gravity; objects moving solely under gravity have zero proper acceleration.

Thank you very much for clearing that up.

 

[pervect]

1) And why there is a need to precise these assumptions and approximations for this particular GR equation of the proper acceleration?
I mean, these assumptions must be made even for the classical Newtonian equation.

2) So there is no need to make the weak field assumption?

3) Can we derive the Force equation (as we know it) from the proper acceleration?

F=ma will, in general NOT work in special (or general) relativity. Generally, people learn special relativity (SR) first, as it's much easier than General relativity. If you have an application other than wanting to know how much a standard test mass would weigh on a scale held by a hoovering observer, you need to be more specific about your question. Otherwise it's fairlly likely there will be some miscommunication and confusion.

Example: a 1kg test mass on Earth would weigh 9.8 Newtons, because one Earth gravity is 9.8 m/s^2.

 

[Ibix]

1) And why there is a need to precise these assumptions and approximations for this particular GR equation of the proper acceleration?
I mean, these assumptions must be made even for the classical Newtonian equation.

You asked if the equation you quoted was correct for a hovering object in GR. The answer is no in general, and in fact it isn't always possible to define "hovering", because if the geometry is time varying it can be impossible to find a sensible definition of "staying in the same place" - let alone calculate the acceleration necessary to hover. But in the case of the Schwarzschild geometry you can do, and the correct formula is the one you gave.

2) So there is no need to make the weak field assumption?

The formula is exact for the specific case of the vacuum around an uncharged non-rotating spherically symmetric body if all other masses (yourself included) are negligible. In other cases it's an approximation, or wildly wrong, or "not even wrong", depending on how badly you violate those assumptions.

3) Can we derive the Force equation (as we know it) from the proper acceleration?

Sure. But there are caveats here too. If you are measuring the rocket thrust needed to hover, measured locally, then you just multiply by the mass. If you are suspending the mass from a string then the force needed at the top of the string is different.

I have difficulty to understand why we can't give a simple definition of the distance R in GR.

Because the geometry is non-Euclidean and the distance across a circle isn't necessarily its circumference divided by $2\pi$. In the case of a black hole the geometry is so far from Euclidean that there isn't a center of the circle - $r=0$ turns out to have more in common with a moment in time than a place in space, and there's no way to pass through it, even hypothetically. So we have to define a "radius" that isn't actually a distance in terms of the circumference of a circle (or the area of a sphere) at that radius.

I don't think all of this was clear in the early days of GR. Perhaps if it were we would define the Schwarzschild circumference as $4\pi GM/c^2$ and not use "Schwarzschild radius" at all.

We can measure the real distance between the center of the distant galaxy and any star in the system. Am I wrong?

It depends what you mean by "real distance", what measurement method you use and, in some cases, how distant. Measuring a distance to the surface of the Sun, or even its center, is simple enough in principle. But that distance will not be the distance round our orbit divided by $2\pi$ (even if our orbit were circular). The difference is tiny, but it's there.

The distance to another star in our galaxy or a nearby one gets a bit trickier to define because they aren't stationary. If you bounce a radar pulse off one (which is possible in principle) you'd get a different answer from what you'd get if you used a really long ruler, because the two methods respond differently to the changing geometry. You can calculate what either method will tell you and the results will differ by far less than typical measurement uncertainties, but it rather undermines the concept of a "real distance" to a star. It gets even more fun when talking about distances to distant galaxies, since there's the metric expansion of the universe to contend with - rulers don't necessarily work and radar doesn't either if you go far enough (the echo never returns, even in principle unless the universe is closed).

The fundamental truth is that a lot of "obvious" stuff about how you can measure things and standard relationships like $C=2\pi r$ rely on assuming Euclidean geometry in a static spacetime, and that is only an approximate model of reality.

We are fairly well into "you need to study GR properly to understand this" territory here. Can I ask why you asked your original question? There isn't a general answer to it (more precisely, the general answer is $U^\mu\nabla_\mu U^\nu$, but that probably isn't helpful), so understanding what you are trying to do might enable us to give specific advice.

 

[PAllen]

And slow speed, and for the case of an isolated system. So still limited in application compared to GR in general.

I have given references in other threads that higher order PPN equations are "unreasonably effective" (a term used by Clifford Will), and have been used very successfully to predict gravitational wave forms and orbital decay for inspiralling BH at high speeds. Numerical relativity is better, but PPN is still used as cross check on numerical relativity up until the final stages of merger.

For the purposes of getting to one order of correction to solar system orbital mechanics, taking into account all GR effects, you have the Einstein-Infeld-Hoffman equations, which are basically first order PPN. These are the ones used in all modern solar system prediction software for astronomy:

https://en.wikipedia.org/wiki/Einstein–Infeld–Hoffmann_equations

Note the first term is Newtonian, all others are 'solved' PPN corrections. These were derived in 1938, but not used for solar system calculations until around 1990 (if my memory serves). Note also that these equations already include non-linear effects, in that the acceleration of one body is dependent on both the acceleration and velocity of all other bodies. That is, compared to EM, there are acceleration dependent 'forces' as well as velocity dependent 'forces'.

 

[sha1000]

You asked if the equation you quoted was correct for a hovering object in GR. The answer is no in general, and in fact it isn't always possible to define "hovering", because if the geometry is time varying it can be impossible to find a sensible definition of "staying in the same place" - let alone calculate the acceleration necessary to hover. But in the case of the Schwarzschild geometry you can do, and the correct formula is the one you gave.

The formula is exact for the specific case of the vacuum around an uncharged non-rotating spherically symmetric body if all other masses (yourself included) are negligible. In other cases it's an approximation, or wildly wrong, or "not even wrong", depending on how badly you violate those assumptions.

Sure. But there are caveats here too. If you are measuring the rocket thrust needed to hover, measured locally, then you just multiply by the mass. If you are suspending the mass from a string then the force needed at the top of the string is different.

Because the geometry is non-Euclidean and the distance across a circle isn't necessarily its circumference divided by $2\pi$. In the case of a black hole the geometry is so far from Euclidean that there isn't a center of the circle - $r=0$ turns out to have more in common with a moment in time than a place in space, and there's no way to pass through it, even hypothetically. So we have to define a "radius" that isn't actually a distance in terms of the circumference of a circle (or the area of a sphere) at that radius.

I don't think all of this was clear in the early days of GR. Perhaps if it were we would define the Schwarzschild circumference as $4\pi GM/c^2$ and not use "Schwarzschild radius" at all.

It depends what you mean by "real distance", what measurement method you use and, in some cases, how distant. Measuring a distance to the surface of the Sun, or even its center, is simple enough in principle. But that distance will not be the distance round our orbit divided by $2\pi$ (even if our orbit were circular). The difference is tiny, but it's there.

The distance to another star in our galaxy or a nearby one gets a bit trickier to define because they aren't stationary. If you bounce a radar pulse off one (which is possible in principle) you'd get a different answer from what you'd get if you used a really long ruler, because the two methods respond differently to the changing geometry. You can calculate what either method will tell you and the results will differ by far less than typical measurement uncertainties, but it rather undermines the concept of a "real distance" to a star. It gets even more fun when talking about distances to distant galaxies, since there's the metric expansion of the universe to contend with - rulers don't necessarily work and radar doesn't either if you go far enough (the echo never returns, even in principle unless the universe is closed).

The fundamental truth is that a lot of "obvious" stuff about how you can measure things and standard relationships like $C=2\pi r$ rely on assuming Euclidean geometry in a static spacetime, and that is only an approximate model of reality.

We are fairly well into "you need to study GR properly to understand this" territory here. Can I ask why you asked your original question? There isn't a general answer to it (more precisely, the general answer is $U^\mu\nabla_\mu U^\nu$, but that probably isn't helpful), so understanding what you are trying to do might enable us to give specific advice.

Thank you again for the such well constructed reply.

1) Initially I was interested if GR could provide a simple force equation F(R). Even though gravitational force does not really exist in GR. But I thought that it was still possible to derive something similar to it, at least locally from the point of view of our perception.

2) I suppose we must use the relativistic Newtonian second law to calculate the force from the proper acceleration: (γ^3)*ma = F

 

[anuttarasammyak]

1) Initially I was interested if GR could provide a simple force equation F(R). Even though gravitational force does not really exist in GR. But I thought that it was still possible to derive something similar to it, at least locally from the point of view of our perception.

Force equation of F(R) in Newtonian mechanics satisfies Poisson equation,
$$\triangle \phi = 4\pi G \rho$$
and gravity potential $\phi$ is approximately
$$g_{00}=1+2\phi$$
in Newton approximation of GR. Thus GR assures Newton' law as an approximation.

 

[PAllen]

2) I suppose we must use the relativistic Newtonian second law to calculate the force from the proper acceleration: (γ^3)*ma = F

For a body not acted on by any non-gravitational forces, there is no proper acceleration at all. Please see my prior post for the minimal generalization of Newton's gravitational law, expressed in similar terms, that approximates well for e.g. solar system mechanics.

Also, your force equation is only valid for force directed parallel to velocity.

 

[sha1000]

For a body not acted on by any non-gravitational forces, there is no proper acceleration at all. Please see my prior post for the minimal generalization of Newton's gravitational law, expressed in similar terms, that approximates well for e.g. solar system mechanics.

Also, your force equation is only valid for force directed parallel to velocity.

I was talking about the proper acceleration of a hovering object which is discussed in previous answers by Ibix and others.

a_p = $\frac{GM}{r^2\sqrt{1-2GM/r}}$

So now the question is if we can take this proper acceleration and put it into the equation
F = γ³ma_p ?

 

[Orodruin]

I was talking about the proper acceleration of a hovering object which is discussed in previous answers by Ibix and others.

a_p = $\frac{GM}{r^2\sqrt{1-2GM/r}}$

So now the question is if we can take this proper acceleration and put it into the equation
F = γ³ma_p ?

That is not the gravitational force. There is no gravitational force in GR. It is the force needed to be provided by other means to keep the object stationary.

Also note that you make explicit reference to a gamma factor. It is unclear what you intend this gamma factor to be.

 

[PeterDonis]

the question is if we can take this proper acceleration and put it into the equation
F = γ³ma_p ?

No, you can't, because that equation is only valid in an inertial frame, and the frame in which the quantities in the proper acceleration formula are defined is not an inertial frame.

 

[PAllen]

I was talking about the proper acceleration of a hovering object which is discussed in previous answers by Ibix and others.

a_p = $\frac{GM}{r^2\sqrt{1-2GM/r}}$

So now the question is if we can take this proper acceleration and put it into the equation
F = γ³ma_p ?

Ok, but originally you seemed to be asking about the simplest useful approximate generalization of Newton's law of gravitation that capture GR effects. Einstein-Infeld-Hoffman equations are exactly that, containing Newtons law of gravitation as the first term.

As for relating force needed to maintain hovering to proper acceleration of a hovering body, note that in a momentarily comoving local inertial frame (in this case, a momentarily stationary free fall frame coinciding with some hovering body), you have that coordinate acceleration equals proper acceleration, and also that simply F=ma is true (m and F measured in the local, momentarily stationary, free fall frame). Recall, as well, that proper acceleration magnitude is a scalar invariant, thus it can be computed in non-inertial Schwarzschild coordinates, then used as described in a local momentaritly stationary free fall frame.

 

[sha1000]

That is not the gravitational force. There is no gravitational force in GR. It is the force needed to be provided by other means to keep the object stationary.

Also note that you make explicit reference to a gamma factor. It is unclear what you intend this gamma factor to be.

Yes, you are right. This is indeed a force needed to keep the object stationary.

A.T. who indicated the proper acceleration equation also wrote: "The force needed to hover it would be the proper acceleration times its mass." F = ma

But the relativistic Newtonian second law is F=dP/dT = F = γ³ma.

So I thought that it would be more convenient to use this relativistic equation.

 

[PeterDonis]

the relativistic Newtonian second law is F=dP/dT = F = γ^3*ma.

This is not a fully general law. It is only valid in an inertial frame, for acceleration parallel to velocity. Neither of those applies to the situation in which you are trying to apply it.

 

[PAllen]

The general form of F=ma in special relativity, using 3-vectors in an inertial frame is:
$\mathbf F = m\gamma^3(\mathbf v \cdot \mathbf a)\mathbf v + m\gamma \mathbf a$

 

[sha1000]

This is not a fully general law. It is only valid in an inertial frame, for acceleration parallel to velocity. Neither of those applies to the situation in which you are trying to apply it.

PAllen wrote: " As for relating force needed to maintain hovering to proper acceleration of a hovering body, note that in a momentarily comoving local inertial frame (in this case, a momentarily stationary free fall frame coinciding with some hovering body), you have that coordinate acceleration equals proper acceleration..."

I'm really not an expert. Does this mean that we can define a co-moving local inertial frame (stationary free fall frame coinciding with hovering body)? From this point would it be possible to apply F = γ³ma?
Since we are dealing with co-moving local inertial frame?