Gravitational Field Transformations Under Boosted Velocity

[Sciencemaster]

TL;DR Summary

Let's say we have a massive body with an isometric gravitational field around it, described by the Schwarzschild metric. How would the field around it be different for a moving observer far away from the field?

Let's say we have some observer in some curved spacetime, and we have another observer moving relative to them with some velocity $v$ that is a significant fraction of $c$. How would coordinates in this curved spacetime change between the two reference frames?

For example, imagine a massive body with an isometric gravitational field around it, described by the Schwarzschild metric. How would the field around it be different for a moving observer far away from the field? Would the field undergo length contraction, as with a Lorentz boost? Or is there some other transformation law we have to apply to curved spacetimes?

I know that changes in coordinate position transform differently between flat and curved spacetimes (i.e. Lorentz Transformations don't apply if you change position in a gravitational field), so I'm curious as to how gravitational fields change with motion. My intuition tells me that a gravitational field would experience length contraction and the like, as the spacetime far away from the body should behave approximately as if it was flat, and celestial bodies with gravitational fields still show evidence of relativistic effects.

So just how would coordinates in a curved spacetime change for an observer with relative motion?

 

[jbriggs444]

The metric is an invariant property of the space time. It is the same for all observers.

If expressed in coordinate form, the formulas may appear different for different coordinate systems. And perhaps that is what you are trying to get at. But it is still the same identical metric no matter how it is formulated.

Curvature is also an invariant property that can be computed at any event in a space time. It derives from the metric. So it does not matter what coordinates you use -- cartesian rest coordinates corresponding to one observer or cartesian rest coordinates corresponding to another. The local space time curvature will be identical for both.

However, perhaps none of that is what you have in mind...

Perhaps you wish to consider something like a solar system with a one solar mass black hole in the center replacing the sun. You wish to view this solar system using something that is (asymptotically at least) flat Minkowski space time. Against this backdrop, the black hole is moving and you wish to know about the spatial shape of this solar system according to these coordinates -- is the solar system length contracted in the direction of its motion?

Yes, I would expect it to be.

 

[Ibix]

How would the field around it be different for a moving observer far away from the field?

There is no general prescription for defining coordinates in a curved spacetime, so there is no general transformation rule.

My intuition tells me that a gravitational field would experience length contraction and the like, as the spacetime far away from the body should behave approximately as if it was flat, and celestial bodies with gravitational fields still show evidence of relativistic effects.

Broadly. The limit of a Schwarzschild metric seen at lightspeed is the Aichelburg-Sexl Ultraboost.

 

[Sciencemaster]

Broadly. The limit of a Schwarzschild metric seen at lightspeed is the Aichelburg-Sexl Ultraboost.

Thank you for the response. However, that link is broken. I found what you were talking about, but you should probably remove the "ml" before the "wikipedia" in the link so that future readers can use it.

I'm having trouble understanding how the delta function works in the ultraboost. Do you just perform the operation $\delta(u)log(r)=log(u)$? I'm trying to understand this metric and the "shape" of its field.
I would imagine it's roughly an ovoid that is contracted in the direction of motion, as a solar system with roughly circular orbits (in a reference frame where it is stationary) whizzing by in a reference frame where it is moving very quickly has the distances between planets and the star contracted in the direction of motion, yet the planets don't immediately get sucked into the star due to being very close to it (relative to the frame where it is stationary), but I'm not certain.

 

[Ibix]

I found what you were talking about, but you should probably remove the "ml" before the "wikipedia" in the link so that future readers can use it.

Indeed - I edited the URL to remove Wikipedia's mobile version indicator, messed it up, and didn't spot it. Now corrected, thanks.

I'm having trouble understanding how the delta function works in the ultraboost.

It means that the gravitational field does look like an impulsive gravitational wave from the point of view of a fast moving observer. Assuming you are approaching in the ecliptic plane there is no "depth" to the orbits of planets. An asteroid belt would be just a straight line.

A way to look at this is to consider the experience of a fast moving object approaching a black hole. It might take seconds, hours, or days of Schwarzschild coordinate time for the object to cross through the region where gravity is non-negligible, but at high enough speed this is negligible proper time for the object. So it must experience the field as a near-zero time phenomenon that causes an abrupt change of direction.

 

[Sciencemaster]

It means that the gravitational field does look like an impulsive gravitational wave from the point of view of a fast moving observer. Assuming you are approaching in the ecliptic plane there is no "depth" to the orbits of planets. An asteroid belt would be just a straight line.

A way to look at this is to consider the experience of a fast moving object approaching a black hole. It might take seconds, hours, or days of Schwarzschild coordinate time for the object to cross through the region where gravity is non-negligible, but at high enough speed this is negligible proper time for the object. So it must experience the field as a near-zero time phenomenon that causes an abrupt change of direction.

Oh, I understand! That's a pretty good analogy! So, if I'm understanding correctly, the gravitational field can be interpreted as kind of a circular flat plane perpendicular to the direction of motion of an observer moving at lightspeed, where the observer will only feel a force if it is on this plane (which is where the delta function comes in). This can be seen in the celestial body's reference frame as the observer experiencing extreme time dilation as it flies by the body, and thus only experiencing the gravitational field for an instant before it either flies by or collides with the body, just like how an observer close to a black hole will experience the force from the black hole in very little proper time! I will note that the reason the fast moving observer experiences an impulsive force is due to its high velocity, and the reason for it happening in your black hole example is the extreme gravitational field, but the fact of the matter is that they can both be seen as time dilation in the object's frame, so the comparison is valid.

One more question, is there such a boost for an observer that's not moving at lightspeed, or perhaps in a different metric (like the de Sitter metric)? If possible, I'd like to try and get an idea for how gravitational fields experience boosts when this special case isn't applicable, if at all possible.

 

[pervect]

TL;DR Summary: Let's say we have a massive body with an isometric gravitational field around it, described by the Schwarzschild metric. How would the field around it be different for a moving observer far away from the field?

Let's say we have some observer in some curved spacetime, and we have another observer moving relative to them with some velocity $v$ that is a significant fraction of $c$. How would coordinates in this curved spacetime change between the two reference frames?

For example, imagine a massive body with an isometric gravitational field around it, described by the Schwarzschild metric. How would the field around it be different for a moving observer far away from the field? Would the field undergo length contraction, as with a Lorentz boost? Or is there some other transformation law we have to apply to curved spacetimes?

I know that changes in coordinate position transform differently between flat and curved spacetimes (i.e. Lorentz Transformations don't apply if you change position in a gravitational field), so I'm curious as to how gravitational fields change with motion. My intuition tells me that a gravitational field would experience length contraction and the like, as the spacetime far away from the body should behave approximately as if it was flat, and celestial bodies with gravitational fields still show evidence of relativistic effects.

So just how would coordinates in a curved spacetime change for an observer with relative motion?

IT's rather technical, but I wrote something about this quite a while back. See the PF thread https://www.physicsforums.com/threa...-tidal-forces-on-moving-objects-redux.511492/

It involves some approximations. One of which is the ability to ignore time dilation. It also makes use of the equivalence of tidal gravity to the Riemann tensor, the mathematical construct that GR actually uses. The main point is that if we could integrate the tidal forces out to infinity, we can recover the "force" at infinity. The presence of gravitational time dilation complicates this - a key assumption is that this can be ignored.

The basic result is that the "force" increases a little faster than $\gamma^2, \gamma$ being $1 / \sqrt{1-(v/c)^2}$. However, the actual result incorporates an additional factor of $\frac{1 + (v/c)^2}{2}$ which I argue is due to spatial curvature. Spatial curvature is ambiguous by itslef, as I'm almost sure people will point out. So I'll add that the original post contains the specific set of basis vectors that I use to define precisely what I mean by that term. IT's a coordinate dependent entity, because it's a piece of a tensor and not the whole tensor. However, the full post has the details necessary to formally disambiguate what I'm referring to.

This result has not been checked by anyone else as far as I know, and it's definitely not peer reviewed.

I believe somewhere I worked out the analogous case where the "force" of gravity was on a block sliding on the floor of an accelerating elevator. This is simpler than the case in an actual gravitatioanl field.

Looking this up, it looks like one of the times I talked about this involved superman holding a weight from a spring while flying at relativistc velocities. See https://www.physicsforums.com/threa...elativistic-speeds.948690/page-2#post-6008592. The calculations are again rather technical, though simpler than the gravitational case - it uses Rindler coordinates, and basically calculates the proper acceleration of Superman as he flies at a constant Z value in Rindler coordinates. This is probably better than the sliding block one I also did.The dimensional argument is that time dilates by a factor of $\gamma$, and because $d^2 x/ dt^2$ involves the square of time, we'd expect an acceleration and force dilates by a factor of roughly $\gamma^2$. This argument isn't really good enough to stand on it's own, but it shows that the result insn't surprising.

 

[pervect]

I think I should add a cautionary remark about interpreting relative acceleration due to spatial curvature as a force. I don't believe that such components transform in the same way a 'force' does. An example will help.

Suppose you have two ships sailing on geodesics on an idealized spherical planet. They travel great-circle routes. They start out at the same point, but using the mathematics of curvature (or possibly other simple arguments can do in a pinch), one can argue that they appear to accelerate twoardds each other. They stop separating when they've travelled 1/4 of the circumference, and they meet again at the antipodes after making a half-circuit of the spherical planet. All we really have here is the most basic example of spatial curvature - the spatial curvature of the surface of a Euclidean sphere.

If we want to use the mathematics of curvature to compute this, we of course use the geodesic deviation, which gives the relative acceleration between the two geodesics as $R^a{}_{bcd} u^b x^c u^d$, u^b and u^d being the velocities of the ships, and x^c being the separation vector between them. So the geodesic deviation equation gives us the relative accerlation between the two geodesics (ships).

Is it logical to call the relative acceleration between the two ships , sailing on the curved planet, a "force"? Is there really a 'force' attracting them? Personally, I don't usually call this a force, except in my last post, where I sort of did. But I'm thinking better of it. Though perhaps one can make a point for calling it a force, see below.

The argument here as applied to the original problem is that electrogravitc part of the tensor can be interpreted as a tidal force, and (by still ignoring time dilation) we can integrate this tidal force to get the "force". Basically, 2m/r^3 , the associated electrogravitic part of the Riemann, integrates to m/r^2. And we thus have our "gravitational force". We do have to ignore time dilation to perform the integration, of course, so it's an approximation at best.

But the full tensor gives more components to the acceleration. In addition to the electrogravitic part of the Riemann, there's a topogravitic part, which I have described as arising from spatial curvature ( none of the associated components of the Riemann in this part of the decomposition involve time). There's a velocity squared dependence here, which we also see in the sailing ships - the relative acceleration between the ships is proportional to their sailing velocity squared (as we can see from the geodesic deviation equation), and so is this mysterious "extra" component in our relative acceleration.

As with the ships sailing on the ocean, it's rather problematical to say that the spatial curvature generates a "force".

But I suppose you can sort-of interpret the relative acceleration due to spatial curvature (of the ships on the ocean, or in the case of the flyby of a massive object) as a force if you really want to - but you wind up with a force that is quadratic in velocity.

This particular problem has no magnetogravitc component to the Riemann - such components would be linear in velocity. The electrogravitic part has no velocity dependence, the magnetogravitic part as a linear velocity dependence, and the topogravitc part has a velocity^2 dependence.

That might actually be an argument in favor of calling the topogravitc part a force to, if one compares it to magnetism - and we do call magnetism a force. In the end, I suppose, ti's a matter of definitions.

 

[Sciencemaster]

IT's rather technical, but I wrote something about this quite a while back. See the PF thread https://www.physicsforums.com/threa...-tidal-forces-on-moving-objects-redux.511492/

It involves some approximations. One of which is the ability to ignore time dilation. It also makes use of the equivalence of tidal gravity to the Riemann tensor, the mathematical construct that GR actually uses. The main point is that if we could integrate the tidal forces out to infinity, we can recover the "force" at infinity. The presence of gravitational time dilation complicates this - a key assumption is that this can be ignored.

The basic result is that the "force" increases a little faster than $\gamma^2, \gamma$ being $1 / \sqrt{1-(v/c)^2}$. However, the actual result incorporates an additional factor of $\frac{1 + (v/c)^2}{2}$ which I argue is due to spatial curvature. Spatial curvature is ambiguous by itslef, as I'm almost sure people will point out. So I'll add that the original post contains the specific set of basis vectors that I use to define precisely what I mean by that term. IT's a coordinate dependent entity, because it's a piece of a tensor and not the whole tensor. However, the full post has the details necessary to formally disambiguate what I'm referring to.

This result has not been checked by anyone else as far as I know, and it's definitely not peer reviewed.

I believe somewhere I worked out the analogous case where the "force" of gravity was on a block sliding on the floor of an accelerating elevator. This is simpler than the case in an actual gravitatioanl field.

Looking this up, it looks like one of the times I talked about this involved superman holding a weight from a spring while flying at relativistc velocities. See https://www.physicsforums.com/threa...elativistic-speeds.948690/page-2#post-6008592. The calculations are again rather technical, though simpler than the gravitational case - it uses Rindler coordinates, and basically calculates the proper acceleration of Superman as he flies at a constant Z value in Rindler coordinates. This is probably better than the sliding block one I also did.The dimensional argument is that time dilates by a factor of $\gamma$, and because $d^2 x/ dt^2$ involves the square of time, we'd expect an acceleration and force dilates by a factor of roughly $\gamma^2$. This argument isn't really good enough to stand on it's own, but it shows that the result insn't surprising.

I see. So, in simple terms, as an observer moves past a body with some velocity, the acceleration it experiences in the direction of the body increases by a factor of $\gamma^2$ relative to the reference frame of the body. This fits with the Aichelburg-Sexl Ultraboost from earlier, as the observer experiences much more force in the instant it moves past the body--a special case of your thought experiment where v=c. This leaves me wondering about how the force changes once the observer moves past the body...
I would imagine that the acceleration acting on the observer would decrease by a factor of about $\gamma^2$ times the distance it has moved past the body relative to the acceleration it experiences in the body's reference frame, or in other words, the acceleration of the observer falls off faster as it moves past the body in the observer's reference frame. This is effectively what I was trying to describe when I was originally asking if a gravitational field could experience length contraction for a moving observer--the moving observer has to be closer to the source of the gravitational field to experience the same amount of acceleration. Would this be the case?

 

[pervect]

My computation, as well as being approximate, is coordinate specific. It more or less has to be, because force is not actually a tensor in GR. And the coordinate system of the analysis I did is that of a static observer, not a moving observer. The Aichelberg-sexl ultraboost would be a different coordinate choice and my analysis would not be directly comparable.

The analysis in flat space-time, on Einstein's elevator, is much less problematic and more straightforwards.

The relative acceleration between two nearby geodesics is well defined in a coordinate independent manner, using the geodesic deviation equation. Assembling these relative accelerations into a "force" requires assuming a specific coordinate system, a specific "chain" of observers, and a specific path to perform the integration. (Plus, along with all these assumptions, I made some simplifying approximations).

It is interesting to compare the gravitational case to the much simpler electromagnetic case, the electric field of a moving point charge. See for instance https://www.physicsforums.com/threa...arge-according-to-purcell.909149/post-5736070.

Another poster, Bill K, tersley commented on the extra factor of gamma for the gravitational case as compared to the electromagnetic case in https://www.physicsforums.com/threads/field-of-moving-point-charge.563012/

 

[pervect]

Oh yes - something else that's useful, which has the advantage that it's published, and peer reviewed. Rather than trying to deal with forces-on-a-distant object, which are problematical because they're not covariant, one can look instead at the effects of a flyby on a swarm of test particles, to see what motion is imparted to the particles in the swarm by the flyby. The space-time is perfectly flat before and after the flyby, which removes all the issues involved with dealing with curved space-time. One can then compare the effects on the trajectories (the total velocity induced by the flyby) on the field of test particles.

This basically looks, not at the force as a function of time, but rather at the total impulse delivered, the change in momentum which is the integral of force * time.

The paper is by Olson & Guarino, https://aapt.scitation.org/doi/abs/10.1119/1.14280, "Measuring the active gravitational mass of a moving object".

If you can get a full copy of the paper (there are some paywall issues), it also has a brief discussion of some of the various concepts of mass (there are several!) in use in General Relativity, which are different from the particular approach Olson & Guarino use in this paper. The fact that there are several different concepts in use and not just one might serve as a gentle warning of the complexities that arise.

The main results of the flyby are summarized in the abstract, however, which is not paywalled.

If a heavy object with rest mass M moves past you with a velocity comparable to the speed of light, you will be attracted gravitationally towards its path as though it had an increased mass. If the relativistic increase in active gravitational mass is measured by the transverse (and longitudinal) velocities which such a moving mass induces in test particles initially at rest near its path, then we find, with this definition, that M_rel=γ(1+β^2)M. Therefore, in the ultrarelativistic limit, the active gravitational mass of a moving body, measured in this way, is not γM but is approximately 2γM.

Again, you see multiplicative factor (in this case 2) in addition to a factor of gamma when you compare the gravitational case to the electrostatic case.

The issues of exactly what happens during the flyby are complicated by the lack of covariance, but Olson and Guarino's approach sidesteps by this by looking instead at the total deflection, basically the total impulse delivered during the flyby.

This again ties in with the comments made by Bill K and my own analysis. If you use the electrostatic case as a comparison, there's a multiplicative factor of gamma. And there's a further multiplicative factor on top of that, Olson and Guarnio's method gives this additional factor as 2.

 

[Sciencemaster]

Alright, that's interesting! Thank you for your responses! One more thing I would like to know, the results you have shared mostly deal with the effects of velocity on acceleration of particles in the direction perpendicular to the motion of the mass. This is helpful, but I'm also wondering about how velocity of a massive object affects acceleration of test particles in the direction parallel to its motion and how quickly this acceleration decreases as the distance from the moving mass increases.

 

[vanhees71]

The paper is by Olson & Guarino, https://aapt.scitation.org/doi/abs/10.1119/1.14280, "Measuring the active gravitational mass of a moving object".

This argues with the socalled "relativistic mass" concept, which is very bad. What it in fact shows is that the source of gravitational fields is not the mass of matter only but any kind of energy, momentum, and stress as predicted by GR.

 

[Vanadium 50]

"How do gravitational fields transform under a boost in velocity" has a simple answer: they don;t. This mixes a Newtonian concept, gravitational fields, with some but not all of relativity. To do this consistently is why Einstein had to throw away the concept of Newtonian gravity and invent GR

Is it possible to kind of partially stitch relativity onto Newton, like epicycles? Probably. But it will only work sometimes, and worse, you will only know when it's not working by comparing with GR. I would say this path leads nowhere.

 

[pervect]

This argues with the socalled "relativistic mass" concept, which is very bad. What it in fact shows is that the source of gravitational fields is not the mass of matter only but any kind of energy, momentum, and stress as predicted by GR.

To me, it shows that relativistic mass is not a bad first approximation as to what causes gravity, but it's inaccurate by a factor of about 2:1. Not great, but to give people without enough background as to at least some small guide as to what going on, it's a start.

An accuracy of 2:1 is not great, the effects one are ignoring are the same mangiutde as the effects one is using in the explanation, but it's at least the right order of magnitude. To get the complete answer, one really do need the stress-energy tensor. The stress energy tensor is an organized collection of energy density, momentum density , and pressure. All of the components have influences on gravity, because they are components of the stress-energy-tensor. Note than an alternate way of describing what I (and others, such as Baez) call pressure is stress - "pressure" here is the same thing as the classical stress tensor, which explains the name of the stress-energy tensor.

Thus, the internal forces that keep a bridge from collapsing can be analyzed in terms of what some call "pressure" and others call "stress". There are unfortunately some subtle differences between the engineering treatment of stress and the GR treatment of the related concept. I actually find the GR concept to be easier to understand, as well as more general. But that's a personal preference, really. Pressure - or stress - whichever name you want to call it, is one of the contributors to the stress-energy tensor and hence a source of gravity.

[add-afterthought]
Another useful tidbit. Generally, matter is so weak that most stress terms are negligible in their gravitational effects, though they do theoretically h ave an effect. The major exception is simply the pressure at the center of a sufficiently massive body, say, a neutron star. (I don't have any numbers on how big it is, but it's probably not negligible). Since large bodies are generally spherical (due to the aforementioned weakness of matter), we can usually get away with a single number for an isotropic pressure, which should be fairly familiar even to someone who is not familiar with the mathematical treatment of stress in it's full generality. FOr those who are familiar, see the "perfect fluid" approximation in GR.