Does a gravitational field have mass?

[duordi]

I ask this question because of conclusions from the following example.

Two masses orbit one another.
The masses are 5 minutes apart at light speed.
The objects are identical and always face each other ( they rotate once per orbit )
The Gravitational force of mass 1 toward mass 2 is directed toward the instantaneous location of mass 2 and not toward a position it was 5 minutes ago.

This indicates that the gravitational field of mass 2 is not regenerated from mass 2 at each new location of mass 2 as this would cause a time delay of the position of mass 2's gravitational field.

Instead the gravitational field of mass 2 orbits mass 1 as an independent object.
That is to say mass 2 and the gravitational field of mass 2 have similar orbits.

This would center the gravitational field of mass 2 at the center of mass 2 without time delay.

For this to occur the gravitational field of mass 2 must have mass itself, however slight.

I know that light can only be attributed mass when it is in motion, however a standing electromagnetic wave can have the appearance of having a constant location and yet the waves creating the constant field are themselves in constant motion.

A second related question is does a standing electromagnetic wave have rest mass?
If I consider the standing wave to be at rest ( not moving or changing field strength. )

Thank you in advance for your response.

Duane

 

[Jonathan Scott]

In both electromagnetism and a simple relativistic approximation to gravity, the force between moving objects includes a velocity-dependent term as well as a static term, and it effectively points towards the extrapolated position where the source would be at the current time, given its position and the way it was moving at the latest time that it could be seen via a light-speed signal.

In electromagnetism, this effect is of course seen as being due to the magnetic field. In gravity, there is a similar effect, which is part of what is usually called "gravitomagnetism", but it is so weak as to be extremely difficult to measure, so it is usually ignored for all practical purposes.

The question of whether there is energy in the gravitational field is not really related to this effect. In relativistic gravity, it depends anyway on your point of view, in that what seems to one person to be a force acting and transferring energy may seem to another to be free fall at constant energy.

A standing wave effectively has rest mass. For a standing wave to exist, the electric and magnetic field strengths must be oscillating, so your assumption "not ... changing field strength" cannot apply, but the overall combined amplitude can remain constant. Any combined system of electromagnetic waves which are not all traveling in the same direction also has some rest mass overall, such as two beams traveling in opposite direction, or light bouncing around within a mirrored cavity.

 

[duordi]

I take that as a "Yes".

I have been reading about the pioneer 10 and 11 anomaly which indicates the existence of an extremely small acceleration toward the sun for which no identifiable source has been discovered at this time.

If the gravitational field of the sun does have a minute amount of rest mass as a standing wave of some form then it could explain the additional acceleration toward the sun.

As the satellites pass through the gravitational field to some distant position more of the total mass of the field would cause an acceleration toward the sun.

It is my understanding that a solid mass inside of a second empty spherical mass will experience no gravitational attraction to the spherical mass.

If this is the case then mass contained in a gravitational field outside of the sphere of the radius of a satellite will have not effect on a satellite orbiting the sun.

If however the satellite were to expand its orbit the suns mass would seem to increase by the amount of the gravitational field mass contained in the sphere of the new satellite orbit radius.

Is there another way to determine what the mass density of a gravitational field might be?

 

[pervect]

In general, there is no consistent way to assign mass to the gravitational field, at least not in the usual sense. For instance, it is possible to assign an energy $(1/2) \epsilon_0 E^2$ to the electric field, there is no analogous formula for "the energy of the gravitational field".

http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html has some information on this

In particular, the principle of local energy conservation was a vexing issue. In the general theory, energy is not conserved locally as it is in classical field theories - Newtonian gravity, electromagnetism, hydrodynamics, etc.

MTW also has some information on the topic in "Gravitation".

So there is no formula that gives the "amount of energy" in the gravitational field.

The issue is even broader - as I think I've mentioned previously, there is not even a single unified defintion of mass in general relativity. Instead there are several definitions, which apply under different circumstances. In some cases, there is no applicable definition, and in such cases there is no known definition for the mass of that particular system.

In some simple systems, several of the defintions may apply. In these cases there is usually an agreement as to the total mass of the system, but no agreement as to "where" that mass is located.

 

[duordi]

Are there any holes in my logic?

The gravitational field energy caused by mass-A at any given point can be assumed to be a constant K multiplied by the "field density".

We know the field density is proportional to the inverse square of the radius from the center of mass.( At least it is very close)

Mathematically integration can be used to compose a total gravitational field energy due to a specific mass.

This formula will contain an unknown constant K of course which converts field density to an energy value.

We can also compare the field densities of different masses as they can also be calculated. Each computation will contain the unknown constant K and vary proportionally to the mass of the object in question.

Now consider a mass object with a gravitational field.
We can define the gravitational field energy equation with an unknown constant K.

Now suppose we separate the original mass into two equal masses at a infinite distance from each other.
We can calculate the energy added to the system to separate the masses.
We also know the mass of each half.
We can calculate equations for the gravitational field energy for both the separate masses and the combined mass which will contain the unknown constant K.

If I assume conservation of energy

The energy added to separate the system must be equal to the total gravitational energy change as the total mass of the mass objects has not changed.

It would not be hard to solve this equation for the constant K, and derive an equation equating a gravitational field to an (energy or mass) quantity .

Are there any holes in my logic?

Also, would GR already take this into account somehow, causing the Pioneer 10/11 scientests to have considered and dismissed this idea as a solution?


PS thanks for the reference Pervect..
It may take me a while to totally digest this information so I apologize if my response does not recognize the full impact of Noether’s work.

 

[duordi]

Based on the reference given. It looks like there is no conservation of mass/energy in GR.

I hate GR!

Although it may work pervect ( pun intended as a complement ) it does not lend itself to the answers I am looking for.

Is there a way that SR can be used in this example, or under what constraints can it be used ?

 

[subhrajit]

I'm not an expert, but can make a few points:
1. "The Gravitational force of mass 1 toward mass 2 is directed toward the instantaneous location of mass 2 and not toward a position it was 5 minutes ago." - the statement is probably not correct. Gravitational fields travel at the speed of light. So the effect is not 'instantaneous'.
2. It is known that energy and mass gives rise to gravitational fields. But as far as I know, gravitational fields are frame independent (since energy & mass are together frame independent). I guess there is no concept of associating mass to a gravitational field.

 

[pervect]

Based on the reference given. It looks like there is no conservation of mass/energy in GR.

I hate GR!

Energy conservation in GR is tricky and an advanced topic. There are some conservation principles that always work, but they basically apply only to pieces of space-time of zero volume. And there are other conservation principles that don't always work, but that do work in important special cases.

As far as popularizations go, for this issue I'd suggest reading (or-rereading)

Is energy conserved in General Relativity?, which I'll quote in part.

Is Energy Conserved in General Relativity?

In special cases, yes. In general -- it depends on what you mean by "energy", and what you mean by "conserved".

In flat spacetime (the backdrop for special relativity) you can phrase energy conservation in two ways: as a differential equation, or as an equation involving integrals (gory details below). The two formulations are mathematically equivalent. But when you try to generalize this to curved spacetimes (the arena for general relativity) this equivalence breaks down. The differential form extends with nary a hiccup; not so the integral form.

The differential form says, loosely speaking, that no energy is created in any infinitesimal piece of spacetime. The integral form says the same for a finite-sized piece. (This may remind you of the "divergence" and "flux" forms of Gauss's law in electrostatics, or the equation of continuity in fluid dynamics. Hold on to that thought!)

 

[pervect]

I'm not an expert, but can make a few points:
1. "The Gravitational force of mass 1 toward mass 2 is directed toward the instantaneous location of mass 2 and not toward a position it was 5 minutes ago." - the statement is probably not correct. Gravitational fields travel at the speed of light. So the effect is not 'instantaneous'.

Careful. You might want to take a look at, for instance, This sci.physics.faq

Does Gravity Travel at the Speed of Light?

To begin with, the speed of gravity has not been measured directly in the laboratory--the gravitational interaction is too weak, and such an experiment is beyond present technological capabilities. The "speed of gravity" must therefore be deduced from astronomical observations, and the answer depends on what model of gravity one uses to describe those observations.

In the simple Newtonian model, gravity propagates instantaneously: the force exerted by a massive object points directly toward that object's present position. For example, even though the Sun is 500 light seconds from the Earth, Newtonian gravity describes a force on Earth directed towards the Sun's position "now," not its position 500 seconds ago. Putting a "light travel delay" (technically called "retardation") into Newtonian gravity would make orbits unstable, leading to predictions that clearly contradict Solar System observations.

In general relativity, on the other hand, gravity propagates at the speed of light; that is, the motion of a massive object creates a distortion in the curvature of spacetime that moves outward at light speed. This might seem to contradict the Solar System observations described above, but remember that general relativity is conceptually very different from Newtonian gravity, so a direct comparison is not so simple. Strictly speaking, gravity is not a "force" in general relativity, and a description in terms of speed and direction can be tricky. For weak fields, though, one can describe the theory in a sort of Newtonian language. In that case, one finds that the "force" in GR is not quite central--it does not point directly towards the source of the gravitational field--and that it depends on velocity as well as position. The net result is that the effect of propagation delay is almost exactly cancelled, and general relativity very nearly reproduces the Newtonian result.

This cancellation may seem less strange if one notes that a similar effect occurs in electromagnetism. If a charged particle is moving at a constant velocity, it exerts a force that points toward its present position, not its retarded position, even though electromagnetic interactions certainly move at the speed of light.

The bottom line: except for very small effects due to gravitational radiation, the force of gravity on the Earth points towards the current position of the sun, not the retarded position.

This has been discussed before, one thread is https://www.physicsforums.com/showthread.php?t=149379

there are probably others.