Is Mass Equivalence Explained by the Relationship Between Energy and Gravity?

[maline]

In formulating GR, Einstein was motivated by the otherwise strange coincidence of the equivalence of gravitational and inertial mass. He removed this by discovering that trajectories determined by gravitation are actually inertial. However, gravity's source is still the tensor T, which is closely related to mass. Isn't this the same coincidence?

 

[mfb]

It is not the same.
In general, you can distinguish between "active gravitational mass" (objects that attract other objects), "passive gravitational mass" (objects that get attracted) and inertial mass.

In Newtonian gravity, all three could be different if you do not care about other conservation laws: $m_i a = \frac{GM_a m_p}{r^2}$ where i=inertial, a=active, p=passive.
It is a very reasonable assumption that active and passive gravitational mass are the same - otherwise physics gets really weird (violation of energy conservation and so on). There is no special reason why this should be identical to inertial mass, however, especially if you compare it to electromagnetism where this equivalence does not exist (electric charges correspond to gravitational mass there).
General relativity requires inertial and passive gravitational mass to be the same.

 

[maline]

Yes, I understand all that. But the question of why M(a) is the same as M(i) still seems to demand a more ontological answer than "otherwise we would have violations of CoE". In fact, GR seems to make this worse: Newtonian gravity is a force interaction between the bodies, so it is intuitive that it depends on both masses symmetricallly. In GR, active and passive gravitational mass seem unrelated!

 

[atyy]

Usually in GR, equivalence principles are linked to minimal coupling. But it is tricky. There is discussion in

http://arxiv.org/abs/0707.2748

http://arxiv.org/abs/gr-qc/0505128

 

[martinbn]

Usually in GR, equivalence principles are linked to minimal coupling. But it is tricky. There is discussion in

http://arxiv.org/abs/0707.2748

http://arxiv.org/abs/gr-qc/0505128

What is the formulation of the equivalence principle in terms of minimal coupling, or do I have to read these papers? By the way, when you say usually in GR, what do you mean, in most standard texts or something else?

 

[PAllen]

Yes, I understand all that. But the question of why M(a) is the same as M(i) still seems to demand a more ontological answer than "otherwise we would have violations of CoE". In fact, GR seems to make this worse: Newtonian gravity is a force interaction between the bodies, so it is intuitive that it depends on both masses symmetricallly. In GR, active and passive gravitational mass seem unrelated!

What GR via the POE does is make M(i) and M(p) the same (really the same - they cannot be separated). Then, the question is between M(p) and M(a). This equivalence is less fundamental, especially in GR. To motivated it, I don't see anything better than conservation laws being true locally.

 

[atyy]

What is the formulation of the equivalence principle in terms of minimal coupling, or do I have to read these papers? By the way, when you say usually in GR, what do you mean, in most standard texts or something else?

I cannot remember but it's something like one should write the Hilbert action then add the matter action, and the matter action contains the fields and their first derivative, and the metric, but no derivatives of the metric. This is found in Hawking and Ellis, section 3.3.

 

[martinbn]

I cannot remember but it's something like one should write the Hilbert action then add the matter action, and the matter action contains the fields and their first derivative, and the metric, but no derivatives of the metric. This is found in Hawking and Ellis, section 3.3.

I still don't understand the connection with the equivalence principle. There is nothing about this in Hawking & Ellis, section 3.3. That section explains how to get the stress energy tensor of a field from the Lagrangian of the field. The following section has a remark that Einstein's equations can be obtained from a variation of a curtain action, but nothing more.

 

[maline]

Come to think of it, I've read that pressure, along with mass, is a source of gravity. So if two bodies of equal mass but different pressure are interacting, won't that result in a a violation of CoE?I'll bet the answer to this is relevant to my first Q.

 

[atyy]

I still don't understand the connection with the equivalence principle. There is nothing about this in Hawking & Ellis, section 3.3. That section explains how to get the stress energy tensor of a field from the Lagrangian of the field. The following section has a remark that Einstein's equations can be obtained from a variation of a curtain action, but nothing more.

Would you accept comma goes to semicolon as an expression of an equivalence principle?

 

[martinbn]

Would you accept comma goes to semicolon as an expression of an equivalence principle?

It is not about me being stubborn and not accepting things. I am just trying to understand the connection between EP and minimal coupling that you mentioned. Now you mention something else. Would you elaborate on it?

 

[atyy]

It is not about me being stubborn and not accepting things. I am just trying to understand the connection between EP and minimal coupling that you mentioned. Now you mention something else. Would you elaborate on it?

The minimal coupling is comma goes to semicolon, and the form in Hawking and Ellis is the Lagrangian form of it.

 

[martinbn]

The minimal coupling is comma goes to semicolon, and the form in Hawking and Ellis is the Lagrangian form of it.

But what is the connection with EP? I reread that part of Hawking and Ellis and still don't see it.

 

[atyy]

But what is the connection with EP? I reread that part of Hawking and Ellis and still don't see it.

The EP means that locally we can use Minkowski coordinates, and the "laws of physics that don't couple to curvature" will be the same as in flat spacetime.

So for the EP to hold, there have to be some laws that don't couple to curvature. If we take the Lagrangian as the fundamental expression of the laws, then that means the fundamental laws don't couple to curvature. The derived laws can couple to curvature.

 

[martinbn]

I think I see what you mean. I suppose I misunderstood you, I thought you were talking about a formulation of the EP. I would say the minimal coupling (comma to semicolon) is a condition on the theory for the theory to satisfy the equivalence principle. The reference to H&E is not really relevant.

 

[atyy]

I think I see what you mean. I suppose I misunderstood you, I thought you were talking about a formulation of the EP. I would say the minimal coupling (comma to semicolon) is a condition on the theory for the theory to satisfy the equivalence principle. The reference to H&E is not really relevant.

Yes, it's just the closest "standard reference" I knew about for the Lagrangian form of comma goes to semicolon, which is how the EP is satisfied within GR. So for example, if one had curvature coupling in the Lagrangian, then it is possible for the EP to fail. The tricky part is that sometimes even though minimal coupling fails, there is a change of variables so that the EP holds, which is what the papers I linked to discuss.

 

[PeterDonis]

I've read that pressure, along with mass, is a source of gravity.

The correct statement is that pressure, along with energy density, is a source of gravity. "Mass" in this context would include a contribution from pressure, as well as energy density.

if two bodies of equal mass but different pressure are interacting, won't that result in a a violation of CoE?

No, because, as above, the "mass" includes whatever contribution there is from pressure. If the two bodies have equal mass but different pressure, they must also have different energy density, so that the contributions from energy density and pressure result in the same mass in both cases.

 

[maline]

Sorry for my ignorance. I just got through elementary SR and we learned that mass is defined as
the magnitude of the energy four-vector for the system. How is that different from the energy density in a CMRF?

 

[PeterDonis]

I just got through elementary SR and we learned that mass is defined as the magnitude of the energy four-vector for the system.

Yes, and in that context, pressure is not a source of gravity because there is no gravity. In SR, spacetime is flat, and the stress-energy tensor (which is the source of gravity) is zero everywhere. If you're talking about pressure as a source of gravity, you're talking about GR, not SR, and that's what I was talking about in my previous post. In GR, "mass" is defined differently (in fact, there is no single definition that always applies; you can only define "mass" in certain kinds of spacetimes), and includes a contribution from pressure as well as energy density.

How is that different from the energy density in a CMRF?

The length of the energy-momentum 4-vector has units of energy, not energy density. You can convert it to an energy density by dividing by the object's volume, but that won't be invariant under Lorentz transformations (unlike the length of the 4-vector, which is).

 

[maline]

Got it. Thanks for clearing that up.

 

[TEFLing]

E² = m² + p²

Energy is related to both mass and momentum. To quote JA Wheeler, "mom-energy".

Is it correct, that the equivalence of inertial mass and gravitational mass is due to all mass being energy and energy being the source of gravity?

 

[maline]

Is it correct, that the equivalence of inertial mass and gravitational mass is due to all mass being energy and energy being the source of gravity?

Yes, of course that's the general idea. [Although you need to look out for the details: not all mass is from energy (some is from pressure) and the stress-energy tensor T, which is the source of gravity, is related more closely to mass than to energy.]
However, "mass is the source of gravity" is true in Newtonian physics as well. Einstein was dissatisfied with the coincidence of having mass produce two different affects, and his GR shows that the way mass responds to gravity is actually an example of inertia. My question is whether the way mass also causes gravity fits in with that level of simplicity.