Do Gravitational Forces Increase for Objects Moving at Light Speed?

[Pratyeka]

Since an object's apparent mass increases as it approaches the speed of light, does it's gravitational forces also increases? (From a stationary observer's point of view)

 

[Ibix]

No. Mass does not increase - relativistic mass does, but the concept has been dropped for decades except in popsci.

There can be no effect of velocity on gravity because velocity is relative. Right now you are doing 99.99999% of light speed with respect to a passing cosmic ray. Do you notice any gravity from yourself?

The gravitational field of a moving object is a different "shape" from a stationary one due to relativistic effects, but it is no stronger.

 

[Pratyeka]

Thanks for the clarification. Relativistic mass is the proper term I was looking for, but I used "apparent mass" instead, which is really not proper.

Any simple way to show how the shape of the gravitational field of a moving object differs from a stationary one?

 

[Orodruin]

Any simple way to show how the shape of the gravitational field of a moving object differs from a stationary one?

Not really. Gravitation does not work like that in relativity. In relativity, gravity is the geometry of spacetime.

 

[Ibix]

Any simple way to show how the shape of the gravitational field of a moving object differs from a stationary one?

Not really because, as Orodruin says, you're trying to visualise a curved 4d structure that isn't even locally Euclidean.

That said, you can make some reasonable statements. The field must be symmetric perpendicular to the direction of motion, and it must be shortened parallel to the direction of motion, at least in the sense that one could build a large sphere enclosing the object at a large distance where spacetime is nearly flat and this sphere must length contract.

Your personal experience of traveling through the field would be a sudden and rather sharp direction change - more sudden and sharper at high speed.

 

[Pratyeka]

Your personal experience of traveling through the field would be a sudden and rather sharp direction change - more sudden and sharper at high speed.

Would it be possible to simulate the orbit of an object moving through this field using a computer, if the programmer understand the math? Has it ever been done?

 

[Mister T]

Since an object's apparent mass increases as it approaches the speed of light, does it's gravitational forces also increases? (From a stationary observer's point of view)

No. There is no apparent mass increase. 100 years ago researchers attributed the behavior of fast moving subatomic particles to an apparent increase in mass, called the relativistic mass. But researchers were already abandoning that notion, attributing the behavior instead to the geometry of spacetime. Unfortunately, textbook authors continued to speak of relativistic mass well into the 1990's. One good reason for removing it was that students thought of it as a genuine generalization of the Newtonian notion of mass, thinking for example as you have that it could be used as a substitute for mass in Newton's Law of Gravitation. Of course it's not that simple. Instead general relativity had to be developed to explain gravitation

 

[Ibix]

Would it be possible to simulate the orbit of an object moving through this field using a computer, if the programmer understand the math? Has it ever been done?

You just use a particle passing near a stationary star and transform the result to a coordinate system where the star is moving. The only difficult bit is arguing about what is a "natural" coordinate system for the second part. There isn't an obvious choice, so the problem is not that we don't have an answer but more that we have no clear winner for the way to present it. You could imagine a 2d array of small spaceships passing through the field and do a "what it looks like" video from inside one of them, but you cannot draw a map of everyone's trajectories.

I'm not aware of anyone making such a video. However, there's a 1985 paper by Olson and Guarino that skips the whole issue of messy coordinates by simply comparing trajectories of test masses long before and long after the interaction with the star. Unfortunately I'm not aware of a freely accessible version of it. What it says is that if you close your eyes during the interaction and only open them when the object is past, the final trajectories look like you passed near a Newtonian object of mass $(1+\frac vc)\gamma M$, where $\gamma=1/\sqrt{1-v^2/c^2}$ and $M$ is the mass and $v$ the velocity of the mass. Note that the "closing your eyes" bit is a fairly major caveat to this - if you keep them open then you will get a lot of clues that gravity isn't increased for moving objects, but rather the field is a different shape (as I said before).

 

[vanhees71]

This is utterly misleading, and it's misleading as almost any argument with "relativistic masses" necessarily are. It's already a confusing mess in SR. In GR it misses even qualitatively the point. The beauty of GR is, among other things, that it unifies the concepts of "inertia" and "gravity", which are different phenomena within Newtonian physics with the equality of "inertial mass" and "gravitational mass" an empirical (and thus astonishing) fact.

Already SR teaches us that the inertia of a body is related to its intrinsic energy (i.e., the energy as measured in the rest frame of its center of energy (sic!)), which is the content of the famous addendum Einstein wrote to his big 1905 paper, containing the equation $E=m c^2$. Einstein himself abandoned the notion of a relativistic mass (or even many relativistic masses) pretty soon and substituted it with the modern notion of invariant mass. See the very nice article by Okun in Physics Today on this:

https://www.docenti.unina.it/webdocenti-be/allegati/materiale-didattico/346435

In GR it turns out that since "inertia" and "gravitational interactions" are unified that the sources of the gravitational field are not the masses but the energy-momentum-stress tensor of matter and radiation.

 

[Ibix]

This is utterly misleading,

I think that's the German translation of "a fairly major caveat".

 

[Pratyeka]

Thank you everyone, that was really informative and definitely corrected my concept of relativistic mass.

 

[pervect]

No. Mass does not increase - relativistic mass does, but the concept has been dropped for decades except in popsci.

There can be no effect of velocity on gravity because velocity is relativee. Right now you are doing 99.99999% of light speed with respect to a passing cosmic ray. Do you notice any gravity from yourself?

The gravitational field of a moving object is a different "shape" from a stationary one due to relativistic effects, but it is no stronger.

I'll have to disagree somewhat with some elements of this statement, though I agree with other elements.

There's a paper that looks at the total velocity induced by a relativistic flyby that I like to use. In the Newtonian analogue, one might say that the velocity change is related to the impulse due to the flyby, the integral of the force over time, as opposed to being related to the force itself. However, the total induced velocity is much better defined, because before and after the flyby, one can use flat space-time as a model, and the velocity before and after the flyby an be compared in their respective flat space-times. IT's also easy to communicate what one is talking about, and physically relevant.

One can imagine a cloud of dust particles, at rest relative to each other, with the self-gravitation of the cloud being not experimentally significant. An object flying through the dustcloud will change the velocities of the test particles in the cloud - one can compare the resulting pertubations for Newtonian gravity and GR.

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

The subject of mass in general relativity is MUCH more complex than this single paper , especially much more complex than the abstract of the paper. Reading the full paper and not just the abstract (if you can get through the paywall somehow) will give one a glimpse of some of the complexities of mass in General relativity. Wiki's article on "Mass in General Relativity" is also not too bad for an overview of the topic. It may not actually be understandable without a graduate level background - but it'll illustrate the complexity, at least.

I digress somewhat - back to my main point. Using relativistic mass actually UNDER-estimates the effects of gravity due to a flyby. The actual induced velocity change by an ultra-relativistic flyby is nearly equivalent to a Newtonian flyby of an object with a Newtonian mass equal to twice the relativistic mass.

Using relativistic mass is wrong, then, by a factor of at least 2:1 as far as giving the total deflection goes. But it's interesting that the actual deflection is larger than a naive model based on Newtonian gravity, Newton's laws, and "relativistic mass".

The factor of 2:1 is also interesting, it also comes up with the deflection of light. I feel there is probalby a relationship there, but that's somewhat speculative on my part.

The Olson paper, as I mentioned, can be considered to be about the impulse, the force integrated over time. The "peak force" is not well defined in GR. Even the peak acceleration is not well defined without other information such as defining a specific coordinate system. However, the gradient of the peak force, i.e. the peak tidal force, the gravity gradient measured with a gravity gradiometer (such as a Forwards mass detector, IS well defined. One can look at how long this tidal force is present, and find that it's present for a shorter time for the relativistic flyby than the Newtonian flyby. So, the peak tidal strain induced by a relativistic flyby would scale as gamma^2, and not just a factor of gamma.

The short summary then. Applying Newtonian formula to GR is just wrong, and it gets the wrong answer. Depending on what one is interested in, it can be wrong by a factor as low as 2:1, depending on what one is interested in. A factor of 2:1 isn't great, but it isn't too awful. The deflection of light is another good example - it's also off by a factor of 2:1. I feel that having some intuition to get an answer that is wrong "only" by a factor of 2:1 has some utility. One may not understand why light defelcts by a factor of 2:1 in GR over Newton's theory without some work, but at least one has the idea that light is deflected by gravity and some idea of how much.

Some care must be taken to keep the error as low as 2:1 though - for instance, the peak tidal force (as measured by a gravity gradiometer, such as Robert Forward's mass detector (see https://en.wikipedia.org/wiki/Robert_L._Forward and look for the entry on his so-called mass detector) would be off by a factor of 2*gamma, where gamma is presumed to be a large number. Understanding why Newtonian results are off by a factor of 2 for either light deflection or for the velocity induced by a relativistic flyby will most likely take a significant amount of work and serious study and I don't have a way of doing the topic justice in a short post aimed at an I-level or below background.

 

[PeterDonis]

The factor of 2:1 is also interesting, it also comes up with the deflection of light. I feel there is probalby a relationship there

There is indeed, since the deflection of light case, for this particular scenario, is just the limit as $v \to c$ of the ultrarelativistic flyby case. (Note that such a limiting process is not always valid; but it does happen to work for this scenario.)

 

[vanhees71]

The subject of mass in general relativity is MUCH more complex than this single paper , especially much more complex than the abstract of the paper. Reading the full paper and not just the abstract (if you can get through the paywall somehow) will give one a glimpse of some of the complexities of mass in General relativity. Wiki's article on "Mass in General Relativity" is also not too bad for an overview of the topic. It may not actually be understandable without a graduate level background - but it'll illustrate the complexity, at least.

"Relativistic mass" doesn't make sense in special relativity already. It's just superfluous and overcomplicating things. There is one kind of mass, and that's the scalar invariant mass. It's given by $P_{\mu} P^{\mu}=m^2 c^2$, where $P^{\mu}$ is the total four-momentum of the system under consideration.

In general relativity "relativistic mass" cannot be interpreted in any way as a physical quantity. It's even hard to define what it might be in the first place. The source of the gravitational field in GR is not mass or mass density (as it is in Newtonian gravitation theory) but the energy-momentum-stress tensor of the "matter fields" (including the electromagnetic field).

 

[pervect]

"Relativistic mass" doesn't make sense in special relativity already. It's just superfluous and overcomplicating things. There is one kind of mass, and that's the scalar invariant mass. It's given by $P_{\mu} P^{\mu}=m^2 c^2$, where $P^{\mu}$ is the total four-momentum of the system under consideration.

I agree with most of this - invariant mass is a scalar, and relativistic mass is not - relativisitic mass is just another name for energy. The sticking point, perhaps, is the remarks about physical significance. More on my thoughts here below.

The idea that a quantity that is part of a tensor (but not the complete tensor neatly wrapped up in a package) has absolutely no physical significance whatsoever seems to me to be a bit of an over-reach. For instance, I have not seen people posting that "An electrical field has no physical significance". The analogy here is that the electric field is a piece of a tensor (the Faraday tensor), just as energy (the other name for relativistic mass) is a piece of a tensor, the energy momentum tensor. Do we really want to insist to everyone that electric fields "have no physical significance"? If we don't, why do we say that energy (aka relativistic mass) has "no physical significance"?

If we add a little bit to the statement, to say that "relativistic mass has physical meaning only in the context of an agreed upon frame of reference" or "the electric field has physical meaning only in the context of an agreed upon frame of reference", I would have no objections.

This point is more important in the context of forums such as PF, where we have a mix of readers of various backgrounds. In an A-level discussion, it is usually pretty clear what people mean when they complain about the lack physical significance. It's still not actually precise, but it's well enough understood what is meant that it's hardly worth a long argument.

In general relativity "relativistic mass" cannot be interpreted in any way as a physical quantity. It's even hard to define what it might be in the first place. The source of the gravitational field in GR is not mass or mass density (as it is in Newtonian gravitation theory) but the energy-momentum-stress tensor of the "matter fields" (including the electromagnetic field).

The current incarnation of the wiki I think points out the difficulties of finding an agreed on concept of mass in general relativity. People are still actively looking for quasi-local mass (Wiki has the references, I'm not personally familiar with their work), and we do have a bunch of different existing concepts of mass in general relativity already, such as ADM, Bondi, and Komar mass. The fact that we have at least three of them and people are still looking for more should serve as an indicator that none of them are totally satisfactory, even without going into all the details. And again, going into all these details requires A-level thread, IMO.

 

[PeterDonis]

The idea that a quantity that is part of a tensor (but not the complete tensor neatly wrapped up in a package) has absolutely no physical significance whatsoever seems to me to be a bit of an over-reach.

I think one thing that is often overlooked here is that many of these "part of a tensor" quantities can actually be given invariant representations. For example, the "energy density" is the $00$ component of the stress-energy tensor; that's a "piece of a tensor", not a complete tensor. But we can express "the energy density as measured by an observer with 4-velocity $u^a$" as $T_{ab} u^a u^b$, which is a scalar invariant, even though it also happens to be numerically the same as "the $00$ component of the stress-energy tensor in the chosen observer's rest frame". So it's not so much that parts of a tensor have "no physical significance whatsoever", as that you have to be careful when defining exactly what physical significance they do have. Any valid definition will end up "bottoming out" in an invariant.

 

[vanhees71]

I think one thing that is often overlooked here is that many of these "part of a tensor" quantities can actually be given invariant representations. For example, the "energy density" is the $00$ component of the stress-energy tensor; that's a "piece of a tensor", not a complete tensor. But we can express "the energy density as measured by an observer with 4-velocity $u^a$" as $T_{ab} u^a u^b$, which is a scalar invariant, even though it also happens to be numerically the same as "the $00$ component of the stress-energy tensor in the chosen observer's rest frame". So it's not so much that parts of a tensor have "no physical significance whatsoever", as that you have to be careful when defining exactly what physical significance they do have. Any valid definition will end up "bottoming out" in an invariant.

In such cases one sees that it's really a curse that physicists usually talk about tensor components but sloppily say tensor.

"The $00$ component of the stress-energy tensor" indeed is the energy density as measured by an observer at rest wrt. the basis of the reference frame the tensor components are referring to, and indeed you can write it in a manifest invariant way by introducing the (frame independent!) four-velocity of the observer. It's "the same as the $00$ component of the stress-energy tensor" only in a basis defining the rest frame of the observer. Indeed, if you have defined a reference frame in terms of a concrete physical situation, you always can formulate all observables in terms of invariants/scalars.