# Equivalence principle

Gold Member

## Main Question or Discussion Point

Hey, so I have a question.

The equivalence principle, the way it has always been taught to me, states that the "gravitational mass" is equal to the "inertial mass". Or, in other words, that the amount of inertia an object has really in some way "equal" (or proportional) to the amount of gravity it exerts.

This seems to me to have some trouble when we consider that energy, not just mass, also warps space-time (since it comes in in the stress-energy tensor, right). So, from my naive (non-rigorous GR training) perspective, the gravity exerted by a box full of super high-energy photons (say coated on the inside with a perfectly reflecting surface so that the photons would be trapped) would not be proportional to its inertia since light has no rest mass and therefore no effective inertia.

It seems to me that for the case of light, the "inertial mass" is zero (hence it travels at the speed of light), but the "gravitational mass" is some small, but finite number (in that the energy it has would warp space-time).

What is the flaw in my reasoning?

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Dale
Mentor
It seems to me that for the case of light, the "inertial mass" is zero (hence it travels at the speed of light), but the "gravitational mass" is some small, but finite number (in that the energy it has would warp space-time).

What is the flaw in my reasoning?
The flaw is that an individual photon has no invariant mass, but a system of two or more photons may have invariant mass. E.g. consider the anhilation of a positron and an electron, per the conservation of the four-momentum the invariant mass of the system is conserved, therefore the resulting system of two photons have the same invariant mass as original positron-electron system.

tiny-tim
Homework Helper
Hi Matterwave!

You're confusing inertial mass with rest-mass.

A particle with rest-mass m0 has inertial mass m0/√(1 - v2/c2) …

this is the same as the energy.

A photon (with zero rest-mass) similarly has inertial mass equal to its energy (depends on the frequency, of course).

"Inertial mass" is the answer to how hard do we have to push it.

A bag of red photons is more difficult to push, and so has more inertial mass, than a bag of blue photons.

The flaw is that an individual photon has no invariant mass, but a system of two or more photons may have invariant mass. E.g. consider the anhilation of a positron and an electron, per the conservation of the four-momentum the invariant mass of the system is conserved, therefore the resulting system of two photons have the same invariant mass as original positron-electron system.
yes this is a very important fact .. The total invariant(static) mass of a system does not equal the sum of invariant masses inside the system.
The total invariant mass of the system equals the sum of energy of the particles(not including the kinetic energy of the system as whole)
If you compress an string , you will increase its invariant mass even though the sum of invariant masses is not affected.

PAllen
2019 Award
Hey, so I have a question.

The equivalence principle, the way it has always been taught to me, states that the "gravitational mass" is equal to the "inertial mass". Or, in other words, that the amount of inertia an object has really in some way "equal" (or proportional) to the amount of gravity it exerts.

This seems to me to have some trouble when we consider that energy, not just mass, also warps space-time (since it comes in in the stress-energy tensor, right). So, from my naive (non-rigorous GR training) perspective, the gravity exerted by a box full of super high-energy photons (say coated on the inside with a perfectly reflecting surface so that the photons would be trapped) would not be proportional to its inertia since light has no rest mass and therefore no effective inertia.

It seems to me that for the case of light, the "inertial mass" is zero (hence it travels at the speed of light), but the "gravitational mass" is some small, but finite number (in that the energy it has would warp space-time).

What is the flaw in my reasoning?
Also, this is an unusual statement of the principle of equivalence. It is normally stated the a body's *response* to gravity is determined by the same mass as its resistance to acceleration. One could posit a full blown principle of equivalence in a universe where there is a fixed background gravitational field, and matter and energy produce no gravitation at all.

Also, this is an unusual statement of the principle of equivalence. It is normally stated the a body's *response* to gravity is determined by the same mass as its resistance to acceleration.
Do you agree that the gravitational field strength produced by a moving body must increase, due to the increase in inertia owing to the relative motion? If it does, would this not be a problem with whether a moving star has the required mass to collapse into a black hole versus a neutron star?

GrayGhost

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PAllen
2019 Award
Do you agree that the gravitational field strength produced by a moving body must increase, due to the increase in inertia owing to the relative motion? If it does, would this not be a problem with whether a moving star has the required mass to collapse into a black hole versus a neutron star?

GrayGhost
Sorry, I don't see any relation at all between these questions and what I said.

Two common formulations of EP are (there are many subtle flavors that we need not get into here):

1) intistinguishability of experiments performed in an accelerating rocket versus in a locally uniform gravitational field. (Einstein)

or

2) intistinguishability of an inertial lab in a gravity free region from an inertial (free falling) lab in a gravitational field. (Will, et. al.)

Neither speeks to mass/energy as a source of gravity. That enters the derivation of GR from carrying over the obvious Newtonian fact that mass is a source of gravity.

PAllen
2019 Award
Do you agree that the gravitational field strength produced by a moving body must increase, due to the increase in inertia owing to the relative motion? If it does, would this not be a problem with whether a moving star has the required mass to collapse into a black hole versus a neutron star?

GrayGhost
To answer this specific, unrelated question, motion of a star as a whole relative to some observer has nothing to do with what collapse history is likely for it. Consider that any answer to this question must be the same as from the star's own frame. It can't collapse in one frame and not in another.

Kinetic energy of consituents in the center of mass frame of the star does indeed contribute to its mass as a source of gravity. It also increases outward pressure. The way these balance is complicated and not something I have any expertise in.

Dale
Mentor
PAllan,

Matterwave basically stated that relativistic mass (or inertia) must equal the produced gravitational field strength. In your response, I did realize that your definition of the equivalence principle was as it's commonly stated.

DaleSpam,

I know that if a star collapses into a black hole, then it must do so per all. So if it does so in the proper frame of the star, then it does for all observers. I've read it many times over the years. I must admit though, I've never quite understood why that is the case, as I am not proficient in the GR model. I assume the gravitational field increases with increased relative motion, yes or no? Here's what I'm wondering ...

Does the gravitational field remain spherical (about a length contracted star) when viewed in luminal motion?

GrayGhost

Dale
Mentor
I know that if a star collapses into a black hole, then it must do so per all. So if it does so in the proper frame of the star, then it does for all observers. I've read it many times over the years. I must admit though, I've never quite understood why that is the case, as I am not proficient in the GR model. I assume the gravitational field increases with increased relative motion, yes or no?
The gravitational field (metric) is a tensor field, so it does not change regardless of the coordinate system used.

The way this works is that gravity does not only couple to mass/energy, but also to momentum and stress. As you increase the velocity of an object its energy does increase, but so does its momentum. Gravity couples to both in a way that ensures that the underlying geometry remains invariant.

Here's what I'm wondering ...

Does the gravitational field remain spherical (about a length contracted star) when viewed in luminal motion?
Well, that is actually a little bit of a loaded question. Considered as a single 4D object, the Schwarzschild spacetime does not have spherical symmetry, it has cylindrical symmetry. You can tilt your coordinate time axis so that it is not parallel to the cylinder axis, but the underlying spacetime will still retain cylindrical symmetry.

However, in 4D just as in 3D, if you slice a cylinder along a plane which is not perpendicular to the axis of the cylinder then you will not get a spherical/circular cross section.