# Test of GR's increase of inertia near bodies?

1. Dec 2, 2012

### harrylin

According to GR, the inertia of bodies is greater near masses.
- https://en.wikisource.org/wiki/Geometry_and_Experience

Has this been tested? I did not find a clear answer, the gravity tests seem to be testing other things. Can it be tested? Intuitively I think that it should be possible, but I don't know how....

2. Dec 2, 2012

### Staff: Mentor

Inertia of bodies measured in which frame? ;)
In your local, free-falling frame, you cannot observe gravity or any of its effects (unless your system is so large that you see tidal effects).

3. Dec 2, 2012

### Staff: Mentor

Before asking whether it can be tested, we need to be clear about what it means. I realize this is Einstein again, and he did say "the general theory of relativity teaches" this, so you're taking it as gospel, but here's how modern GR would rephrase what Einstein said:

The presence of "ponderable masses" (i.e., a nonzero stress-energy tensor) in a spacetime determines what the inertial frames are at any point of that spacetime: in other words, it determines which states of motion are "inertial" (freely falling) and which are not. Since any object with no forces acting on it other than gravity will move on a freely falling worldline, if we take that object at a particular event, and we know its initial velocity at that event, we can predict its entire trajectory just by knowing what "ponderable masses" are around.

Near a large mass, like the Earth, the local inertial frames--i.e., the freely falling states of motion with some given initial velocity--"point in a different direction" than they would in a region with no large mass nearby; the local inertial frames point inward, towards the large mass. That is what Einstein meant by saying that the large mass "increases the inertia" of objects.

On this view, of course, we already have a huge pile of experimental evidence that says that ponderable masses do "increase inertia". But there's more. What if the ponderable mass is rotating? It turns out that GR in this case predicts an *extra* effect of the nearby mass, called "gravitomagnetism", which should, for example, affect the orbital parameters of satellites orbiting a rotating planet, as opposed to a non-rotating one. This has actually been measured:

http://archive.sciencewatch.com/dr/fmf/2011/11janfmf/11janfmfCiuf/

This is significant because it is purely a GR effect; it does not appear at all in Newtonian gravity, and yet it still confirms the idea that the matter in the Earth is determining the local inertial frames near the Earth.

(Btw, Cuifolini, who headed up the research described in the linked interview, wrote a book with John Wheeler called Gravitation and Inertia which goes into great detail about this whole issue of what determines inertia, Mach's Principle, is GR "Machian", and so on. It's a difficult read in many places if you're not familiar with the math, but it does have some good discussion of these issues that can be followed without knowing the mathematical background.)

4. Dec 2, 2012

### harrylin

This is about inertia, not gravity - although it should be the same for both. And of course it must be a non-local test, just as time dilation tests are non-local. But it is still not clear how it can be tested, if at all.

5. Dec 2, 2012

### harrylin

:grumpy: I take nothing as Gospel, and I suspect that Einstein was wrong on this. But your reply suggests that modern GR is gospel for you.
What he meant was his intended Machian effect, according to which inertia is caused by the matter in the universe. I can find nothing Machian in your "reformulation".
Yes, I meant the Lense-Thirring effect in my OP.
Thanks. I understand that GR is not fully Machian, which is the reason for my question.

6. Dec 2, 2012

### Staff: Mentor

If you mean do I think modern GR is an exactly correct physical theory, of course not. It obviously has a limited domain of applicability; it's classical, not quantum, and it predicts curvature singularities, which aren't physically reasonable IMO.

If you mean do I think modern GR is a consistent theory which has been well tested within its domain of applicability, then yes, I do, with some qualifiers because any evidence we have about the interiors of black holes, for example, has to be indirect, so we can't be as sure about it (particular in view of the fact that we don't have a good understanding of the effect of quantum corrections).

You don't consider "the presence of ponderable masses (i.e., a nonzero stress-energy tensor) in a spacetime determines what the inertial frames are at any point of that spacetime" to be Machian? Not even a smidgen?

Remember that I didn't say "nearby ponderable masses"; I just said "ponderable masses". Near the Earth, for example, the Earth accounts partially for what the inertial frames are in its vicinity, but not completely; if you don't add some kind of boundary condition to account for the rest of the matter in the universe, you won't get the right answers. Using the Schwarzschild metric to approximate the inertial frames around the Earth assumes that spacetime far away from the Earth is asymptotically flat; that amounts to assuming that the rest of the matter in the universe is spherically symmetrically distributed around the Earth, since GR predicts that spacetime in a vacuum region inside a spherically symmetric mass distribution is flat. The Earth's effect on inertial frames in its vicinity is then a correction to this flat solution. Gravitomagnetism is just an additional correction due to the Earth's angular momentum.

Ok, but I don't think Einstein, in the lecture you linked to, was just talking about the Lense-Thirring effect. I think he was also talking about what I talked about above, that the effect of a gravitating mass like the Earth on local inertial frames is a correction to the averaged effect of the rest of the matter in the universe.

7. Dec 2, 2012

### pervect

Staff Emeritus
I'm not quite following this exposition. I was reading the section in MTW about Mach's principle and the origin of Inertia (around pg 543)- it seems as if it should be related, but I'm not following that, either. It starts out with an assumption that the universe is closed (no reason that it should be in modern cosmology), and then starts to get unclear

8. Dec 2, 2012

### Staff: Mentor

I agree that the MTW discussion is not very clear; the book by Cuifolini and Wheeler that I referred to is basically a book-length expansion of that discussion that tries to make the issues involved clearer.

As far as I know, the theorem that the metric in a hollow vacuum region inside a spherically symmetric mass distribution is Minkowski does not depend on the spacetime as a whole being spatially closed. So it would be valid in an open universe, and since our current cosmological models (meaning the simple FRW models that we can solve analytically) model the universe as spherically symmetric about every point, that theorem can be applied in the way I stated, to justify the use of an asymptotically flat metric to model the local patch of spacetime around the Earth (or the Solar System). But this amounts to saying that, in order to completely account for what the local inertial frames look like in the Earth's vicinity, it's not enough to just look at the mass of the Earth; you have to take into account all the matter in the universe; the spacetime curvature due to the Earth is really a correction to the "background" metric due to the rest of the matter in the universe (which in our local patch of spacetime looks like a flat Minkowski metric).

Does that help?

9. Dec 2, 2012

### pervect

Staff Emeritus
I can understand frame dragging - but I don't understand what it has to do with the claim that "the inertia of bodies increases near masses".

So my question is - what is this "inertia" that is increasing? What tensor (I presume it's a tensor) represents "inertia" that changes in this manner? How do we transform these abiguous words about "inertia increasing" into nobn-ambiguous math - or better yet, into actual measurement (which seems to be the OP's request).

MTW talks about "York's momentum density of weight 5/3, $\pi^{ab}$. My two (related) issues are that I don't know if this is what Einstein meant, and I don't understand this "momentum density" tensor, either. Of course if point 1 is wrong, then point 2 is irrelevant to this discussion...

10. Dec 2, 2012

### Staff: Mentor

I should have clarified that I can't be sure that what I said is what Einstein meant by "ponderable masses increase the inertia of bodies". I agree that "increase of inertia" is not a good way to describe the effect of a gravitating mass on local inertial frames. But I can't think what else Einstein could have been referring to.

11. Dec 3, 2012

### pervect

Staff Emeritus
I found mention of this in "The Meaning of Relativity" online at http://www.gutenberg.org/ebooks/36276

You'll see the following near pg 106

No tensors were used, just the equations of motion using Christoffel symbols. My take on a modern interpretation would be "gravitational time dilation", not "increase in inertia".

12. Dec 3, 2012

### harrylin

OK... then I cannot explain why you used the word "gospel", as you knew that neither of us takes anything for gospel.
Not in the sense that you phrased it - that is in essence still Newtonian with just a change of meaning of the word "inertia". Even the distant gravitational effect that you mentioned is similar as with Newton's theory. The difference is that in Newton's theory there is as much inertia in a nearly empty universe as in a full universe.
As the passage is much longer than that sentence but short enough to include here, I'll do that now:

The general theory of relativity teaches that the inertia of a given body is greater as there are more ponderable masses in proximity to it; thus it seems very natural to reduce the total effect of inertia of a body to action and reaction between it and the other bodies in the universe, as indeed, ever since Newton's time, gravity has been completely reduced to action and reaction between bodies. From the equations of the general theory of relativity it can be deduced that this total reduction of inertia to reciprocal action between masses — as required by E. Mach, for example — is possible only if the universe is spatially finite.

Last edited: Dec 3, 2012
13. Dec 3, 2012

### harrylin

Interesting, thanks! I'll have a look at that. It looks to me that I was asking about point 1, and that Peter replied about point 2. And just as quick first idea: it sounds plausible that an increase in inertia can be detected as an increase in time dilation.