# I Can we increase an object's gravitational force by adding energy?

#### jms4

Summary
Can we increase an object's gravitational force by adding energy? as energy is related to mass, which is related to gravity?
like a shot bullet or arrow has negligibly more gravitational force than a still bullet or arrow?,
m=e/c^2
F=Gm/r^2,
thus, F=Ge/(c^2*r^2)
where e represents the (mass of the object + energy added to the object)
thus more the energy, more the gravitational force, even though it would be negligible?

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#### Ibix

Summary: Can we increase an object's gravitational force by adding energy? as energy is related to mass, which is related to gravity?

like a shot bullet or arrow has negligibly more gravitational force than a still bullet or arrow?,
No. It's more complex than that. Kinetic energy doesn't increase mass (obviously so, since a moving object is stationary to someone else), and general relativity isn't just Newtonian gravity with a few extra terms plugged in.

However, if you have two masses moving at high speed and they collide, the mass of the composite object (assuming no mass gets ejected) is slightly larger than the sum of the two masses, and this is attributable to the energy. Mass is not an additive quantity in relativity.

#### weirdoguy

1. Energy is related to mass and momentum: $E^2=(mc^2)^2+(c\vec{p})^2$. Energy of a moving bullet is higher than energy of bullet that is standing still, but also it's momentum is now non-zero in a way that it's mass is the same as before.
2. You can't use Newton's gravitational law in relativity. Also gravity is not a force in GR.

#### jbriggs444

Homework Helper
A charged battery or a heated stone has [very very slightly] more mass than an uncharged battery or a refrigerated stone. This mass increase is quite real -- it gravitates and resists acceleration.

#### PeterDonis

Mentor
A charged battery or a heated stone has [very very slightly] more mass than an uncharged battery or a refrigerated stone.
Yes, but the mass that increases here is the invariant mass, not the relativistic mass.

#### 1977ub

No. It's more complex than that. Kinetic energy doesn't increase mass (obviously so, since a moving object is stationary to someone else), and general relativity isn't just Newtonian gravity with a few extra terms plugged in.

However, if you have two masses moving at high speed and they collide, the mass of the composite object (assuming no mass gets ejected) is slightly larger than the sum of the two masses, and this is attributable to the energy. Mass is not an additive quantity in relativity.
I had become convinced that kinetic energy increases the effective gravitational influence of a body - as measured by an observer who finds the body to be moving. No?

#### PeterDonis

Mentor
I had become convinced that kinetic energy increases the effective gravitational influence of a body - as measured by an observer who finds the body to be moving. No?
No.

What governs how a body acts as a source of gravity is the body's stress-energy tensor. In the simple approximation we have been implicitly using in this thread, where we treat the body as a single object with no internal structure, the stress-energy tensor is just the object's invariant mass. The object's invariant mass doesn't change in a frame where the body is moving.

For a bound system composed of multiple objects that might be moving relative to each other, the kinetic energy of the individual objects relative to the system's center of mass does contribute to the overall system's gravitational influence, but that's because it contributes to the overall system's invariant mass.

#### pervect

Staff Emeritus
Summary: Can we increase an object's gravitational force by adding energy? as energy is related to mass, which is related to gravity?

like a shot bullet or arrow has negligibly more gravitational force than a still bullet or arrow?,
m=e/c^2
F=Gm/r^2,
thus, F=Ge/(c^2*r^2)
where e represents the (mass of the object + energy added to the object)
thus more the energy, more the gravitational force, even though it would be negligible?
I believe one of the best answers to the intent of this question is the Olson and Guarino paper, Olson, D.W.; Guarino, R. C. (1985). "Measuring the active gravitational mass of a moving object".

Gravitation in general relativity has aspects that are not well modeled solely by the idea of force. Because you are asking this question in a GR forum, I am assuming you are interested in what General relativity predicts. Because you quote only Newtonian formula, and because you are asking about it in terms of "force", I am assuming that you are not familiar with the relevant formula of General relativity. This is not surprising, it's a complex theory.

An approach that does work, as described in the paper, is to consider a space-time that is basically flat, consisting only of a swarm of test particles that are initially relativity at rest. What happens during the flyby according to GR is hard to describe in familiar Newtonian terms, because the presence of the moving mass distorts the fabric of space-time. However, both before and after the relativistic flyby, space-time is flat and has the familiar spatial geometry. The space-time geometry may also be familiar, it's the flat space-time geometry of special relativity.

We can observe the relative velocities between particles in the cloud of test particles before and after the flyby. When we do so, we find the velocities s of the test particles are perturbed. In particular, there is a change in the relative velocities between the test particles. We can analyze this pertubation in GR and in Newtonian physics, though neither result is likely to be intuitively familiar, not even the Newtonian one.

The result, as described in the abstract of Olson & Guarnio's paper, is:

Olson et al said:
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 Mrel=γ(1+β2)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.
It may be difficult to get a hold of the full text of the Olson & Guranio paper, but the abstract is easy enough to find. The interesting result is that the moving mass has even more impact on the velocities in the cloud of test particles than the relativistic factor of gamma - for an ultra-relativistic flyby, nearly twice as much impact.
The factor of two that we see in the Olson & Guarnio thought experiment is similar to the factor of two we see in the extra deflection of light by massive objects, one of the first experimental tests of General Relativity.

The basic idea of the Olson & Guarino experiment would also approximate such things as the pertubations of orbits of planets in the solar system by the flyby of a relativistic star. This works mainly because gravity in the solar system is weak enough that one can get good approximations of the orbital motions from Newtonian theory.

#### PeterDonis

Mentor
The result, as described in the abstract of Olson & Guarnio's paper, is
The language in which this result is expressed, while valid, is likely to be misleading to the OP. A much plainer way of describing the result is that objects that fly by a gravitating mass at relativistic speeds do not have the trajectories you would expect from a Newtonian calculation using the mass $M$ of the gravitating mass. As the speed of the flyby approaches the speed of light, the bending of the trajectory approaches twice the bending predicted by Newtonian gravity; of course the limiting case of this is the bending of light itself in GR, which is twice the "Newtonian" value predicted by taking the limit of the trajectory of a test particle with nonzero mass as the mass is taken to zero while holding the energy per unit mass fixed.

Another way of describing the result, which shows how it is consistent with what I and others have been saying so far in this thread, is to note that the stress-energy tensor and the spacetime curvature induced by the gravitating mass are the same for the test particles described in the paper, which are moving at relativistic speed relative to the mass, and for another set of test particles which start out at rest relative to the mass. The difference in the trajectories does not arise because the mass, as a source of gravity, or the spacetime geometry it induces, is different for the two sets of test particles. The difference arises because of the difference in the velocities of the two sets of test particles relative to the mass.

#### PeterDonis

Mentor
Btw, a previous thread on a similar topic is here:

My post #15 in that thread basically says the same thing I said in post #9 here just now.

#### Ibix

I had become convinced that kinetic energy increases the effective gravitational influence of a body - as measured by an observer who finds the body to be moving. No?
I think that you have to be very careful with language here. It's certainly the case that a very high speed particle passing through a gravitational field will be curved more than a naive Newtonian analysis would suggest. But I don't think you can interpret that as an increase in effective mass, because the details of the trajectory will not match any Newtonian field. Think of it this way: imagine a sphere around a star, big enough that at its surface spacetime curvature is negligible. Now approach it at near the speed of light - what shape is the sphere? It's going to be Lorentz contracted. You can't apply the Lorentz transforms like that inside the sphere because spacetime is curved, but to a fast moving observer who does not enter the sphere the path of one who did enter would have a very sharp kink in it compared to a Newtonian model that wouldn't forshorten the sphere.

Kinetic energy does enter the stress-energy tensor, but it doesn't look like a straight increase in mass.

#### jms4

No. It's more complex than that. Kinetic energy doesn't increase mass (obviously so, since a moving object is stationary to someone else), and general relativity isn't just Newtonian gravity with a few extra terms plugged in.

However, if you have two masses moving at high speed and they collide, the mass of the composite object (assuming no mass gets ejected) is slightly larger than the sum of the two masses, and this is attributable to the energy. Mass is not an additive quantity in relativity.
Nothing to do with mass here, what I meant is that does energy cause Gravity, like a bullet travelling at 99.99% the speed of light could produce a stronger gravitational force than a stationary one?

#### jms4

I believe one of the best answers to the intent of this question is the Olson and Guarino paper, Olson, D.W.; Guarino, R. C. (1985). "Measuring the active gravitational mass of a moving object".

Gravitation in general relativity has aspects that are not well modeled solely by the idea of force. Because you are asking this question in a GR forum, I am assuming you are interested in what General relativity predicts. Because you quote only Newtonian formula, and because you are asking about it in terms of "force", I am assuming that you are not familiar with the relevant formula of General relativity. This is not surprising, it's a complex theory.

An approach that does work, as described in the paper, is to consider a space-time that is basically flat, consisting only of a swarm of test particles that are initially relativity at rest. What happens during the flyby according to GR is hard to describe in familiar Newtonian terms, because the presence of the moving mass distorts the fabric of space-time. However, both before and after the relativistic flyby, space-time is flat and has the familiar spatial geometry. The space-time geometry may also be familiar, it's the flat space-time geometry of special relativity.

We can observe the relative velocities between particles in the cloud of test particles before and after the flyby. When we do so, we find the velocities s of the test particles are perturbed. In particular, there is a change in the relative velocities between the test particles. We can analyze this pertubation in GR and in Newtonian physics, though neither result is likely to be intuitively familiar, not even the Newtonian one.

The result, as described in the abstract of Olson & Guarnio's paper, is:

It may be difficult to get a hold of the full text of the Olson & Guranio paper, but the abstract is easy enough to find. The interesting result is that the moving mass has even more impact on the velocities in the cloud of test particles than the relativistic factor of gamma - for an ultra-relativistic flyby, nearly twice as much impact.
The factor of two that we see in the Olson & Guarnio thought experiment is similar to the factor of two we see in the extra deflection of light by massive objects, one of the first experimental tests of General Relativity.

The basic idea of the Olson & Guarino experiment would also approximate such things as the pertubations of orbits of planets in the solar system by the flyby of a relativistic star. This works mainly because gravity in the solar system is weak enough that one can get good approximations of the orbital motions from Newtonian theory.
Thanks

#### Nugatory

Mentor
Nothing to do with mass here, what I meant is that does energy cause Gravity, like a bullet travelling at 99.99% the speed of light could produce a stronger gravitational force than a stationary one?
No. One way to see this is to consider that a bullet passing you at .9999c and the bullet at rest while you move past it at .9999c are just two different ways of describing the same physical situation: you and the bullet are moving are moving at .9999c relative to one another. So which one should have a stronger gravitational field?

The total energy of a system does affect its gravitational field, but as some of the posts above suggest, the relationship is a lot more complex than energy causing gravity.

#### Ibix

Nothing to do with mass here, what I meant is that does energy cause Gravity, like a bullet travelling at 99.99% the speed of light could produce a stronger gravitational force than a stationary one?
No. As I said, a moving object is stationary according to another observer so the gravity can't be different. However, other forms of energy such as heat (energy that cannot be made to go away by picking a different frame of reference) does contribute.

#### PeroK

Homework Helper
Gold Member
2018 Award
Nothing to do with mass here, what I meant is that does energy cause Gravity, like a bullet travelling at 99.99% the speed of light could produce a stronger gravitational force than a stationary one?
Consider an object falling to Earth. Then consider that process in a frame of reference where the Earth is moving at high speed.

A) where the motion is perpendicular to the line of the falling object. Owing to time dilation, a naive calculation would show a reduced force of gravity. The object would be measured to fall more slowly in the frame where the earth is moving.

B) where the motion is in the same direction as the falling object. Owing to length contraction and time dilation, the object would be measured to fall even more slowly.

So much for relativistic mass and kinetic energy increasing the gravitational force!

#### PeterDonis

Mentor
what I meant is that does energy cause Gravity, like a bullet travelling at 99.99% the speed of light could produce a stronger gravitational force than a stationary one?
I already answered this in post #7, with some clarifications in post #9 and a link to a previous thread discussing the same question in post #10.

#### pervect

Staff Emeritus
Basically, what happens is that in General Relativity, energy, momentum, and pressure all contribute to gravity. There is a name for an organized collection of these quantities, this name is the stress-energy tensor. So the mathematics is that it's the stress-energy tensor that causes gravity.

So basically, the answer to the question of what happens when you add energy to a system may depend on more details than how much energy you add. You need to look at a fuller description to determine what happens to the momentum and energy of the system. You may additionally have to worry about the pressure distribution in the system, though under some circumstances (an isolated system without significant amounts of gravitational energy, for instance a planet, you typically don't). There are cases where you do need to worry about the pressure distribution, though.

I do agree with Peter that the best way of describing things in general is via invariant quantities. My apologies if the way I have described things has caused any confusion about this point. However, I do have concerns about the invariant approach in this context - in particular, it's wrong to plug the invariant mass of a system into Newton's law of gravity and expect to get sensible results. It's also wrong to put relativistic mass into Newton's law and expect to get sensible results.

#### PeterDonis

Mentor
I do have concerns about the invariant approach in this context - in particular, it's wrong to plug the invariant mass of a system into Newton's law of gravity and expect to get sensible results.
This is not because the invariant approach has issues. It's because Newton's law of gravity is not an invariant equation. Obviously you have to plug an invariant into a properly invariant equation in order to get sensible results.

It's also wrong to put relativistic mass into Newton's law and expect to get sensible results.
Here both the equation and the quantity (relativistic mass) are not invariant so yes, this will not give sensible results.

#### 1977ub

I think that you have to be very careful with language here. It's certainly the case that a very high speed particle passing through a gravitational field will be curved more than a naive Newtonian analysis would suggest.
And it will have an pull on an observer more than Newtonian gravity would predict?

#### pervect

Staff Emeritus
pervect said:
It's also wrong to put relativistic mass into Newton's law and expect to get sensible results.
Here both the equation and the quantity (relativistic mass) are not invariant so yes, this will not give sensible results.
The above option, though, is only off by a factor of 2:1, when suitably averaged. This is the Olsen & Guarino result, with the "averaging" process meaning that impulse (force*time) is averaged over the entire flyby period into a velocity change.

This isn't great, but it's still much better than the other option I mentioned, (which was to plug the invariant mass into Newton's law, which is even more wrong).

Unfortunately, to get correct answers to even such a simple problem, both Newton's laws, and the idea that gravity is a force, are inadequate.

My guess is that most readers are unwilling to abandon the notion that gravity is a force. For such readers, I don't have any better alternatives than option 2 currently, with the addendum that it's off by a factor of 2:1.

Perhaps you have a better idea, but because the concept of gravity as a force is not based on tensors, and thus not in the framework of invariant, geometric, objects, I don't think a really accurate answer can be given without explaining the entire framework in which the answer is given. Unfortunately, a full explanation of this is an A level topic.

I do agree that at the A level, invariant geometric objects are the best tools.

#### Ibix

And it will have an pull on an observer more than Newtonian gravity would predict?
I'm not sure that question is well defined, since gravity doesn't really "pull" in general relativity.

You can say that the acceleration needed to hover at fixed $r$ is higher in GR than in Newton (glossing over differences in exactly what $r$ means), but that's at zero speed relative to the mass. But something doing nearly $c$ will be well above escape velocity for any path where Newton is remotely valid, so it doesn't need to hover.

I don't think you can really compare Newtonian and relativistic concepts like this, with honourable (and very carefully phrased) exceptions like the Olson and Guarino paper that pervect cites.

#### PeterDonis

Mentor
I don't think a really accurate answer can be given without explaining the entire framework in which the answer is given
And if that's the case (and I'm not saying it isn't), then I think the best answer that can be given below the "A" level is to just give the answer without trying to give supporting argument, and then say that the supporting argument requires an "A" level discussion.

#### PeterDonis

Mentor
If you mean #25 plus the related #41, they do not address the (wrong) scenario - adding relativistic "mass" without taking more space - as discussed in #23, which shall point out a (real) paradox of that scenario.
You are continuing to repeat erroneous statements and hijacking another poster's thread. You have now been banned from further posting in this thread.

"Can we increase an object's gravitational force by adding energy?"

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