How does GR not allow for antigravity?

[ranger]

I'm just curious as to how GR doesn't allow this.

--thanks.

 

[Daverz]

In Newtonian gravity the lack of anti-gravity is an accident: we observe that inertial mass is equivalent to gravitational mass to a high degree of accuracy, enough so that we can elevate this to a "weak" principle of equivalence, but the possibility that $m_I \neq m_G$ is not ruled out a priori (love the chance to use that Latin phrase). Try googling Eötvös experiments (you can leave the umlauts off).

In GR, freely falling bodies follow geodesics in spacetime that only depend on initial position, time, and 4-velocity*, so this is a built-in property of spacetime and no longer an accident.

* This is somewhat idealized, I suppose.

 

[Thrice]

Hmm would it violate the equivalence principle?

 

[lduych]

In Newtonian physics, mass is simply an indirect reference to the degree of difficulty (F) involved in changing something's direction of motion or its position of rest. So, to have antigravity either the gravitational constant would have to be negative or you would have to have negative values for 1 or 3 of the following: m1, m2, d1, or d2.

But even antimatter has positive mass, and d1 and d2 are both scalar. And there you have it!

[Don't believe a word of it.]

 

[Daverz]

Hmm would it violate the equivalence principle?

Everything is affected the same way by gravity, so there's only one kind of "gravitational charge". So you can't shield from gravity the way you can with electromagnetic forces where there are 2 opposite kinds of charge.

 

[vanesch]

I'm just curious as to how GR doesn't allow this.

If you would have some "anti-gravity" matter, and some gravity matter, you would be able to distinguish a "falling elevator" frame from a "free frame in outer space", which is disallowed for in GR.
The anti-gravity matter would fall UPWARD in the falling elevator, while the normal matter would "float in the elevator". On the other hand, in the free frame far away, both matter and anti-grav. matter would remain "inertial".
You are, in GR, not supposed to be able to make such a distinction.

 

[JesseM]

If you would have some "anti-gravity" matter, and some gravity matter, you would be able to distinguish a "falling elevator" frame from a "free frame in outer space", which is disallowed for in GR.
The anti-gravity matter would fall UPWARD in the falling elevator, while the normal matter would "float in the elevator". On the other hand, in the free frame far away, both matter and anti-grav. matter would remain "inertial".
You are, in GR, not supposed to be able to make such a distinction.

Are you sure? In Newtonian physics, negative mass would fall towards a positive-mass planet just like positive mass would. And both negative mass and positive mass would be repulsed by a negative-mass planet.

On this thread pervect mentions in post #6 that one of the mouths of a traversable wormhole could eventually become negative in mass, and quotes this article which says that "The mouth with positive mass will attract more mass to it, while its negative-mass twin will gravitationally repel any nearby mass." So it sounds as though negative mass in GR would indeed lead to a kind of antigravity, and since it's allowed by GR I doubt the equivalence principle is violated.

 

[Jheriko]

Are you sure? In Newtonian physics, negative mass would fall towards a positive-mass planet just like positive mass would. And both negative mass and positive mass would be repulsed by a negative-mass planet.

This is not correct. If we were able to assign polarity to the gravitational masses they would behave much like the charges in electromagnetism, except that the rules for attracting and repelling are reversed. We can see this from Newton's law of gravity if we add signs to the quantities $m_1$ and $m_2$:

For both +ve or -ve we get:

$$F = \frac{G{m_1}{m_2}}{{r^2}}$$
or
$$F = \frac{G(-{m_1})(-{m_2})}{{r^2}}$$
which are both the same as
$$F = \frac{G{m_1}{m_2}}{{r^2}}$$

i.e. attractive force

For one or other -ve and the other +ve we get:
$$F = \frac{G(-{m_1}){m_2}}{{r^2}}$$
$$F = \frac{G{m_1}(- {m_2})}{{r^2}}$$
Which are both the same as
$$F = - \frac{G{m_1}{m_2}}{{r^2}}$$

i.e. repulsive force

So, if we allow negative masses we violate the equivalence principle, since we can tell the difference between accelerating in Einstein's elevator and falling in a gravitational field, by using a negative mass and observing the presence of the repulsive force.

 

[lalbatros]

Apparently, it looks like your question is based on an analogy.
Could you explain the analogy you have in mind?
Is it an analogy with charged particles?
Or is it an analogy with matter/antimatter ?

 

[JesseM]

This is not correct. If we were able to assign polarity to the gravitational masses they would behave much like the charges in electromagnetism, except that the rules for attracting and repelling are reversed. We can see this from Newton's law of gravity if we add signs to the quantities $m_1$ and $m_2$:

I don't think you're right about that, the acceleration of a mass $$m_1$$ should depend only on the gravitational "polarity" of the mass $$m_2$$ that it's next to, not on the polarity of $$m_1$$ itself. Consider the equation you posted:

$$F = \frac{G{m_1}{m_2}}{{r^2}}$$

If this represents the force on $$m_1$$, then we have $$F = m_1 a$$, where a is the acceleration of $$m_1$$ in the direction of $$m_2$$. So, substitute that in:

$$m_1 a = \frac{G{m_1}{m_2}}{{r^2}}$$

Then divide both sides by $$m_1$$

$$a = \frac{G{m_2}}{{r^2}}$$

So, the acceleration depends only on $$m_2$$; if $$m_2$$ is positive, $$m_1$$ will accelerate towards it, while if $$m_2$$ is negative, $$m_1$$ will accelerate away from it.

]For both +ve or -ve we get:

$$F = \frac{G{m_1}{m_2}}{{r^2}}$$
or
$$F = \frac{G(-{m_1})(-{m_2})}{{r^2}}$$
which are both the same as
$$F = \frac{G{m_1}{m_2}}{{r^2}}$$

i.e. attractive force

Your equations are correct, but you're forgetting that if both $$m_1$$ and $$m_2$$ are negative, this translates to:

$$-m_1 a = \frac{G{m_1}{m_2}}{{r^2}}$$

This is not an attractive force, because if you divide both sides by $$m_1$$ you get

$$-a = \frac{G{m_1}{m_2}}{{r^2}}$$

or

$$a = - \frac{G{m_1}{m_2}}{{r^2}}$$

So, I still don't see why there should be any violation of the equivalence principle if negative masses were possible.

 

[ranger]

This may sound like something I shouldn't be asking. But I can't really grasp your explanations because the term anti-matter and negative mass make no sense to me. Can someone explain?

 

[Jheriko]

Your equations are correct, but you're forgetting that if both $$m_1$$ and $$m_2$$ are negative, this translates to:

$$-m_1 a = \frac{G{m_1}{m_2}}{{r^2}}$$

This is not an attractive force, because if you divide both sides by $$m_1$$ you get

$$-a = \frac{G{m_1}{m_2}}{{r^2}}$$

or

$$a = - \frac{G{m_1}{m_2}}{{r^2}}$$

So, I still don't see why there should be any violation of the equivalence principle if negative masses were possible.

This makes good sense, I was overconfident since I recognised the parallel with the Coulomb force law, of course the same logic doesn't actually apply since the Coulomb force is not dependent on mass.

If what you say is correct... and I see no reason for it not to be, then my logic on it breaking the equivalence principle is also flawed like you say.

The better argument against negative masses is probably that they result in a negative energy density that persists... but I don't fully understand why that is disallowed yet.

I will research this further, as it is quite interesting...

 

[lalbatros]

First thought:

If a test body is attracted by another massive body,
could we not reverse the sentense,
and say that it is repelled by all the other masses in the universe ?

Second thought:

For electric charges, same signs imply repulsion.
For masses, it is the contrary.
Does not that have (deep) consequences?

Michel

 

[JesseM]

First thought:

If a test body is attracted by another massive body,
could we not reverse the sentense,
and say that it is repelled by all the other masses in the universe ?

I don't think that makes sense, if you assume "all the other masses in the universe" are repelling the test body, there's no reason they should necessarily push it in exactly the direction of the other massive body.

Second thought:

For electric charges, same signs imply repulsion.
For masses, it is the contrary.

Not exactly, two positive masses will attract each other, but two negative masses will repel each other (so even if negative mass existed, it wouldn't naturally tend to accumulate into concentrated bodies like stars and planets).

 

[JesseM]

This may sound like something I shouldn't be asking. But I can't really grasp your explanations because the term anti-matter and negative mass make no sense to me. Can someone explain?

Antimatter doesn't have anything to do with negative mass, it's just like normal matter but with the charges reversed--for example, regular electrons have a negative electrical charge, while the antimatter version of an electron, called a "positron", would have a positive electrical charge, but would be identical to an electron in terms of other properties like mass (although if the electron has a charge in terms of the other forces besides gravity and electromagnetism, namely the strong and weak nuclear forces, I think those would be reversed in the positron as well). I believe antimatter is required to exist by quantum field theories, and anyway it's been found experimentally.

Negative mass, on the other hand, is more hypothetical. The basic idea is just that there are a lot of equations (like the equation for the Newtonian gravitational force $$F = \frac{G m_1 m_2}{r^2}$$) that include a mass variable which can have any positive value--m=10 grams, m=38 kg, whatever--so we can ask what would happen if we plugged a negative value like "m=-80 kg" into those same equations. There isn't any reason to believe negative-mass particles exist, although under relativity negative mass is equivalent to negative energy, and it is thought that negative energy densities can be created by the Casimir effect.

 

[vanesch]

Are you sure? In Newtonian physics, negative mass would fall towards a positive-mass planet just like positive mass would. And both negative mass and positive mass would be repulsed by a negative-mass planet.

Yes, that is why negative mass as such is not "antigravity" stuff by itself. I was talking about "anti-gravity" stuff, which, I took it, must be something which falls upward on the Earth's surface. In other words, something that "goes the other way" as normal matter in a gravitational field. It is *this* which is disallowed for by GR, unless you relax some rather fundamental postulates of it.

I never said that negative mass was anti-gravity stuff. You made this implicit connection...

 

[JesseM]

Yes, that is why negative mass as such is not "antigravity" stuff by itself. I was talking about "anti-gravity" stuff, which, I took it, must be something which falls upward on the Earth's surface. In other words, something that "goes the other way" as normal matter in a gravitational field. It is *this* which is disallowed for by GR, unless you relax some rather fundamental postulates of it.

I never said that negative mass was anti-gravity stuff. You made this implicit connection...

Fair enough, this alternate notion of anti-gravity matter hadn't occurred to me. I think it is reasonable to characterize negative mass as a type of anti-gravity matter though, since even though it would fall downward on earth, its own gravity would have a repulsive effect on other objects...in theory if you had a very dense clump of negative mass, with greater mass than the entire earth, it would appear to fall upward in the Earth's gravitational field, although it would actually be pushing the entire Earth away from it at a greater rate than it was being pulled toward the Earth (pushing the Earth out of its orbit, among other things).

 

[pervect]

While it wouldn't fall up, negative mass objects, if they could be controlled (which turns out to be a rather big if) would still be somewhat useful even though they would fall down.

If you have a 1 million ton ship that you want to get off the ground, for instancem, by adding 999,999 tons of negative mass ballast, you now have a ship that weighs only one ton. If you ballast it closely enough, it will float to the top of the atmosphere like a baloon because it will weigh less than the air it displaces.

However, negative mass is a lot more difficult to control than one might think for thermodynamic reasons. Think about a solid lump of negative mass encountering a larger lump of positive mass. F=ma, and by the principle of equivalence, something with a negative gravitational mass has a negative inertial mass. This means the negative mass tends to move towards any force. So if a negative mass object touches the wall of a larger positive mass container, it will experience a repulsive force, which will cause it to move towards (not away from) the container. The positive mass will also move away from the negative mass, but if the positive mass is larger, its accleration will be less. The result is an untable feedback situation that causes the negative mass to jam itself ever more tightly against the positve mass.

 

[vanesch]

I think it is reasonable to characterize negative mass as a type of anti-gravity matter though, since even though it would fall downward on earth, its own gravity would have a repulsive effect on other objects...

Yes. But in GR, there are lots of strange effects, and even more so if you allow for weird energy-momentum tensors. I would say that everything that, under pure gravitational influence, "follows a geodesic", is still "normal gravitation". The "science-fiction" antigravity stuff doesn't follow geodesics (without using other fields, such as electromagnetism), but "goes in the other way" ; this is what makes it incompatible with GR: because "going the other way" is not a frame-independent concept and hence its world line would not be a geometrical entity in GR (surface of the Earth frame versus falling accelerator frame).

In Newtonian physics, you can allow for genuine anti-gravity stuff if you make the signs of the passive gravitational mass and of the inertial mass opposite.

As a recall, the inertial mass is the m in m.a = F ;
If A and B are two objects, then the force ON A BY B is: F = G m_Ap m_Ba/r^2

where m_Ap is the passive gravitational mass of A and m_Ba is the active gravitational mass of B. The force of A on B is F' = G m_Bp m_Aa / r^2 (in the other direction).

However, the equivalence principle says that for an object, m = m_p = m_a.

 

[jetpaq]

i think its becuse the entire galxy we are in is moving at a speed somwhere below but close to light speed ...then our lil solar system is also moving relative to the speed of our galaxy...and then finally the speed of our planet in rotation ,moving around among all of these enornous inertial forces creates the distortion ,we call gravity ...and it is actually this very distortion that acts upon us rather than this small simple view of gravity...I know i sound nuts ...thats cause i am lololl good luk