Center of Gravity vs Center of Mass

[Grahamshortuk]

Is there anything in General Relativity that forbids the centre of a gravity field of a body of mass being different from the centre of its mass? Might some future technology be able to displace an objects gravity field (or part thereof) such that the mass falls into its own gravity field (placed ahead of it for example) and gains acceleration?

[Jeff Root]

It is *THE* central *ASSUMPTION* of General Relativity that
gravitational mass and inertial mass are the same thing.
From that assumption, plus Special Relativity, General
Relativity follows.

However, the assumption was made in a universe apparently
composed almost entirely of ordinary matter, and before
anything at all was known about "dark matter". Chances are
that dark matter and antimatter behave just like ordinary matter
in this regard, but they haven't been tested yet. Some other
form of matter might exist for which the principle of equivalence
does not apply, but there is no indication of any such matter.

So it seems most unlikely.

-- Jeff, in Minneapolis

[EDG]

I don't think anyone's ever assumed that the universe was composed almost entirely of antimatter! Pretty much all of the 'normal' (baryonic) matter in the universe is well... matter.

[Strange]

I don't think anyone's ever assumed that the universe was composed almost entirely of antimatter! Pretty much all of the 'normal' (baryonic) matter in the universe is well... matter.

Although, as they are indistinguishable, it could be all antimatter.

[Jeff Root]

OOOOOPS!!!!

The first time I said "antimatter", obviously I meant "ordinary matter".
I'm going to go back and fix that...

-- Jeff, in Minneapolis

[Jeff Root]

I'll add that it *is* known that the inertial mass of antimatter
is the same as the inertial mass of ordinary matter. What isn't
known is whether the gravitational masses are the same.
Experiments to measure that using antihydrogen atoms are
planned to be done at CERN within the next few years.

-- Jeff, in Minneapolis

[Nereid]

Is there anything in General Relativity that forbids the centre of a gravity field of a body of mass being different from the centre of its mass? Might some future technology be able to displace an objects gravity field (or part thereof) such that the mass falls into its own gravity field (placed ahead of it for example) and gains acceleration?

I'm not sure quite what you're asking, so I don't really know if what follows is directly pertinent.

In GR, it's mass-energy that is the star player, not just mass itself. In addition, gravitational wave radiation carries energy (that's a simplification).

Further, there are second-order effects which occur in 'the strong field' limit (that's just a fancy way of saying that the mass-energy densities are very high - and the masses great (usually) - such as around a double neutron star in a close orbit, or near a SMBH). It may be possible - in principle - to create a situation somewhat like that you describe in the strong field limit.

Also, I think I introduced you to the Sachs-Wolfe effect, in another thread, didn't I? In some way that somewhat resembles what you're asking, except that it's light which is doing the 'falling'.

Finally, if dark energy turns out to be something other than the cosmological constant (this can be quantified by its equation of state; holler if you'd like more details), some version of your question might be possible, albeit only on a cosmological scale.

[Grey]

I'll add that it *is* known that the inertial mass of antimatter
is the same as the inertial mass of ordinary matter. What isn't
known is whether the gravitational masses are the same.
Experiments to measure that using antihydrogen atoms are
planned to be done at CERN within the next few years.

And even without that direct experimental evidence, there's slightly less direct, but still very strong, evidence that their gravitational masses are the same (with an experimental error of about 0.04%) provided that conservation of energy holds.

[Grahamshortuk]

Interesting. I gotta admit I was kind of expecting a stright "no. The gravity of a mass must have it's cener at the same place as the center of it's mass."
So are we saying that there may be a situation where a mass, say a star or planet or black hole or starship, may on fact NOT have it's centre of gravitational pull at it's actual centre? That it may be possible for it's gravity "well" to be displaced in space from the object itself?

As a side question, would gravity waves be able to phase, with troughs adding to troughs, peaks adding to peaks, and troughs and peaks cancelling out?

[astromark]

Observation and understanding do not support this idea at the present time...
The OP makes a question of interest as to be able to project a mass so as to use it to accelerate is interesting... but as yet fiction. Sorry.
From this I ask a question; A loosely bound mass like a pile of scree with a lot of space inside it would at its center of mass have a very neutral gravity.
From outside of it the gravity of its combined mass is felt... Is that right ?

[Grahamshortuk]

I should expect that relative to the outside environment the gravity should be the same as if it were caused by the sum of the masses of the constituent parts. Inside the collection of objects the gravity would i expect be as if multiple small gravity fields were all pulling in seperate directions towards each part.

[Jeff Root]

Graham,

It is possible that it is possible, though it is more likely that it is
impossible. :-)

Mark,

Yes. So ...

Grey,

What sort of slightly less direct evidence do you refer to?
Is it that gravitational mass of ordinary matter is equivalent
to the total energy content of that matter? Could you say
a little bit about it?

-- Jeff, in Minneapolis

[Nereid]

Interesting. I gotta admit I was kind of expecting a stright "no. The gravity of a mass must have it's cener at the same place as the center of it's mass."
So are we saying that there may be a situation where a mass, say a star or planet or black hole or starship, may on fact NOT have it's centre of gravitational pull at it's actual centre? That it may be possible for it's gravity "well" to be displaced in space from the object itself?

The simple answer, to your straight-forward question, is "No. The gravity of a mass must have its centre at the same place as the centre of its mass. However, this is a rather too simplistic summary; there are many subtleties about this question, if it is being asked within the framework of General Relativity rather than Newtonian mechanics. These subtleties are far too difficult to explain in 140 characters, or less, even if you are a GR maven."

As a side question, would gravity waves be able to phase, with troughs adding to troughs, peaks adding to peaks, and troughs and peaks cancelling out?

No (with some caveats).

[Nereid]

I should expect that relative to the outside environment the gravity should be the same as if it were caused by the sum of the masses of the constituent parts. Inside the collection of objects the gravity would i expect be as if multiple small gravity fields were all pulling in seperate directions towards each part.

It's clear, to me at least (by now), that you are approaching this entirely within the framework of Newtonian mechanics.

Nothing wrong with that ... except that the universe seems to behave according to GR, not Newtonian mechanics (though that is a very good approximation in a quite large region of parameter space).

Within the framework of your statement, it is correct (insofar as any vague, words-only, statement can be).

[Grey]

Grey,

What sort of slightly less direct evidence do you refer to?
Is it that gravitational mass of ordinary matter is equivalent
to the total energy content of that matter? Could you say
a little bit about it?

Imagine the following setup. We take a proton and an antiproton (any particles would do) at the top of a tall tower, and have them annihilate each other, producing a bunch of radiation. We know just how much energy is produced, and from experiment, that's based on the inertial mass and seems unaffected by anything like the height at which the particles annihilate. Then we send that radiation down to the bottom of the tower. We use the energy in the photons to reproduce a proton-antiproton pair, but because of the gravitational blueshift, the photons will have a little bit more energy at the bottom than they did at the top. If conservation of energy holds, then that excess energy has to be exactly the right amount to lift the particles back to the top of the tower. Otherwise, I could run this scenario either backward or forward and create energy out of nothing. We know from experiments (like those studying the Mössbauer effect) that the energy difference for the photon and the proton is the same, to within 0.02%. Therefore, the difference between the proton and the antiproton cannot be more than 0.04%. Take a look here for a little more detail, and some links to references discussing the matter.

[blueshift]

Interesting. I gotta admit I was kind of expecting a stright "no. The gravity of a mass must have it's cener at the same place as the center of it's mass."
So are we saying that there may be a situation where a mass, say a star or planet or black hole or starship, may on fact NOT have it's centre of gravitational pull at it's actual centre?

The way this last sentence is constructed sheds a different view than your OP. The center of the Earth's gravitational pull is not at its actual center. The mass of the Earth does not have uniform density throughout. And the center of gravity of a starship is very unlikely to be located at its "actual" center. Are you confusing "center of mass" with "center of its mass"?

[Jens]

I may be way off, but I think that essentially the center of mass of an object is where the center of gravity is. So there is no way that you could say, displace the center of gravity of a person by 2 meters to the left so that you would be attracted to your own gravity and keep moving faster and faster in that direction. I think that gravity is inherently wedded to mass. For one thing, from the conservation of matter, if you could move the gravity from a person 2 meters to the left, then the person would no longer have mass, which means they could no longer exist.

I think the OP is envisaging a system where you could shift the gravity field of an object outside that object, and then be attracted to your own gravity so that you could gain momentum without any expenditure of energy. I don't think there is any honest way to answer that with anything but no.

Unless you go to Hogwarts, of course. There, it is probably possible.