Questions of the equivalence principle

[Garth]

Being full of doubts I have some questions of the equivalence principle.

1. In the presence of gravitational fields the Einstein
Equivalence Principle (EEP) is a necessary and sufficient condition
for the Principle of Relativity, (PR). Here I summarise PR as the
doctrine of no preferred frames of reference. In the absence of such
fields the EEP becomes meaningless, although then the PR does come
into its own and is appropriate in Special Relativity (SR), which was
formulated for such an idealised case. However, if we now re-
introduce gravitational fields, i.e. gravitating masses, do we not
then find that the PR collapses? For in that case is it not possible
to identify preferred frames of reference? Such frames being those of
the Centre of Mass (CoM) of the system in question and the universe
as a whole, (that in which the Cosmic Microwave Background is
globally isotropic.) The CoM is preferred in the sense that only in
that frame of reference, that is the centroid measured in the frame
co-moving with the massive system, is energy conserved as well as
energy-momentum. But if the PR is not valid in the presence of
gravitational masses then surely the EEP cannot be either?

2. Should not gravitational time dilation apply equally both to
photons and the atoms they interact with? If so whence gravitational
red shift?

3. Should not the total relativistic energy, measured in the
system's CoM, of a freely falling body, be conserved as no work is
being done on or by it?

4. In order for a gravitational theory to be consistent with SR
should not the rest mass of a body include its gravitational
potential energy? In which case questions 2 & 3 are resolved;
gravitational red shift is not caused by the photon losing energy but
by the measuring apparatus gaining it. Yet the EEP forbids it to do
so, for according to the EEP rest mass is invariant.

5. According to the EEP a stationary electron on a laboratory
bench is accelerating w.r.t. the local Lorentzian freely falling
inertial frame of reference. According to Maxwell's theory of
electromagnetism an accelerating electric charge, such as an
electron, radiates. So why doesn't it? Or, if it is thought that such
an electron actually does radiate, what is the source of such
radiated energy? However, note that in the preferred CoM frame of
reference the electron is not accelerating.

6. In the normal expanding cosmological solution of General
Relativity (GR) what exactly is expanding? If it is space-time
itself, as demanded by the theory, then what expands with it? As the
Schwarzschild solution for gravitational orbits is embedded in that
space-time should not its solutions co-expand? Also as the
Bohr/Schrödinger/Dirac equations of atomic physics are also so
embedded then should not their solutions expand? If, as a
consequence, gravitational orbits and atoms together with the
physical rulers constructed of those atoms so co-expand with the
universe, then surely there would be no detectable expansion?
Therefore cosmological red shift cannot be caused by recession, but,
in a similar way to the case of gravitational red shift, it must be
caused by the measuring apparatus, that is all fundamental particles,
secularly gaining inertial mass.

 

[VantagePoint72]

I'm sorry that I can't give a more complete answer to such a well-written question (I'll try and tackle number 1 in this post), but I'll do my best. I'm not entirely sure what you mean by the principle of relativity. Einstein formulated two theories, the special and general theories of relativity. They are both based on the fact that something observed in one frame of reference can be viewed differently but just as validly in another frame of reference...is that what you're referring to? If it is, you would be correct in saying that it prohibits the existence of an absolute frame of reference for UNIFORM motion...uniform motion can be defined only by an objects velocity relative to other objects. Non-uniform motion, however, does have a benchmark: spacetime itself. In the classic example of Newton's bucket, in which the surface of water in a spinning bucket is concave do to lack of centripetal forces on the water that act on the bucket, thereby forcing the water up the side of the bucket. The question is: in an empty Universe, with nothing to define the motion by, would the water's surface still by concave (or more appropriately, would two rocks tied together by a rope spread apart pulling the rope taught.) Relationists would say no, only by its acceleration relative to the average mass distribution of the universe can an object's non-uniform motion be defined. Einstein disagreed, citing spacetime itself as a reference point for accelerated motion. In his universe, the water's surface is still concave in an empty universe. The equivalence principle only says that gravitational attraction and accelerated motion are indistinguishable from one another, and equates the two. The centres of masses of gravitating objects do not serve as a preferred reference point. They serve as a valid one, but not the preferred one. You could define your motion relative to an objects CoM just as you could define it by anything else. The CoM serves as one of many inertial frames of reference, the principle of relativity is not violated, the equivalence principle isn't invalidated, it's simply irrelevant as the CoM is an inertial reference frame, as previously stated. An important thing to note is that there is no theoretical centre of the Universe from which all motion is globally isotropic as it's everywhere: due to Hubble's law, no matter where you are in the Universe, it still appears isotropic as objects farther 'out' are moving out faster than you and objects farther 'in' are moving more slowly than you. The principle of relativity is valid in the presence of masses just as it is when the masses aren't present. Granted, it was this issue that Einstein himself wrestles with and the General Theory of Relativity is the result of it: it shows how the PR applies in gravitataional frames of reference. The equivalent principle is definitely present in the presence of matter. It defines the gravity of the matter and is the reason the PR applies! Gravity doesn't violate any of the basic principles. It only complicates them somewhat. General relativity to the rescue.

 

[Haelfix]

Excellent questions, and very well phrased.. Unfortunately the answers are a little tough to explain, and subtle.

Let me ask you, what lvl of math are you familiar with in GR?

I'll give you quickie responses in english, but I don't think its going to help much without seeing the math

1) The PR is perfectly valid in GR, what is no longer trivial to explain, is what you mean exactly by 'conservation of energy'. Let me be specific, the other 3 forces are perfectly well defined by energy conservation laws in GR, both in terms of stress-energy, self interactions etc. However gravity and itself energy is no longer trivial to explain, partially due to a lack of gauss terms in the tensor equations, and the fact that there exists a mathematical choice of frames and boundary conditions at infinity in the definition. So in certain cases of particular metrics, you sometimes get something well defined, in others (like the FRW) you end up with an intrinsic ambiguity. Google for John Baez and relativity, he has a good summary of this problem.

2) What do you mean by 'interact' (btw I know what you are trying to ask, now I'm asking you to think a little bit about what that entails precisely).

3) Nope, see 1), or ask me again to show it to you explicitly with tensors.

4) I'm glad you see that there is a measurement problem inherent in 2).. However that's not the only problem that makes 3) ill defined... As for the invariant mass.. EEP says a point particles mass is invariant up to second order effects.. Be very careful, it does not say that gravity and acceleration are identical. Now, consider a mass configuration that induces some metric.. I ask you, what frame are you measuring things in, and what are the details of the metric b/c as I mentioned before, its not clear how to define a bodies global energy.

5) Yea.. ok this is the really hard one to explain. It has to do with what you mean by radiation =) The details involve something known as the far field approximation. But let's be clear.. I claim that there are two spacetime points of interest with 2 separate cases (one accelerating, the other gravitating). the spacetime points are the electrons rest frame, the other the measuring devices rest frame.. In one case you will get real radiation, in the other you will get Faraday induction (can you guess which case and frame corresponds to which?). In both cases, the answer of the measuring device will be the same, though the physical mechanism is naively different. What, how do you tell which case is which, and what's this nonsense about different physical mechanisms, doesn't that break the EP? The resolution involves essentially taking a third observer, and bringing him away to infinity.. and that's where things get hazy to explain, without going into mathematics. Radiation (Unruh) is defined in a bit of strange way, but relative to that asymptotic observer, he can distinguish between the gravitational case and the accelerating case. That third observer does violate the EP in a sense, but he is unphysical .

6) No, atomic systems don't expand.. Or at least if they do expand, its so vanishingly small that it makes no difference. The typical example of this is to draw a point on a balloon, or to stick coins on its surface, and blow the balloon up. So yes spacetime is what is expanding, but the coins or the point, do not. Spacetime expansion as governed by the FRW model is a sort of generic average over all metrics, its an astronomically large average, at distance scales many orders of magnitude larger than local effects.

 

[Garth]

Thank you for some intelligent replies!
My answers to the above questions are found in my published theory Self Creation Cosmology, see my thread https://www.physicsforums.com/showthread.php?t=32713 and the papers referred there; published in refereed journals and on the Physics arXiv. You'll be able to make up your own mind as to what level of maths I'm familiar with.

LastOneStanding - I agree about the water bucket, I am a 'Machian' or 'relationist',the question is; is GR? Einstein (and Weinberg) were not sure; in fact Mach's Principle is incompatible with GR because MP singles out a preferred frame of reference, that is locally the Centre of Mass of the system or cosmologically the frame in which the cosmic microwave background radiation is globally isotropic. In fact how can you reference to space-time, it's just empty vacuum with no preferred frame of reference; what do you anchor the coordinate system of that frame to?
What I mean by the principle of relativity is precisely the requirement not to have a preferred frame of reference, which is directly connected to the equivalence principle, yet the equivalence principle only makes sense when there are gravitating masses around and the principle of relativity only makes sense when they are not! (Think about it!)

Haelfix - Consider gravitational red shift. take two identical pieces of apparatus and raise one to the top of a high tower. Send a photon from the bottom apparatus to the other at the top. The photon is observed red shifted. GR interprets this as the photon having lost energy, yet the photon has lost energy even though it has traveled across curved space-time on a null-geodesic instantaneously with no forces acting on it. Meanwhile a lot of work has been done of the top apparatus lifting it and its rest mass has remained the same, its extra gravitational potential energy has dissappeared. Self creation interprets this the other way round. The free photon's energy remains invariant while the apparatus rest mass has increased by the energy required to raise it.
The mathematics is in my published papers, try 'The derivation of the coupling constant in the new Self Creation Cosmology', http://arxiv.org/abs/gr-qc/0302088 .

 

[VantagePoint72]

OK, I think I see what you're saying. As for 'is GR Machian'...well, Einstein wanted it to be. I believe he even sent a letter to Ernst Mach excitedly telling him that the theory of relativity was going to prove the relationist viewpoint correct. Ultimately though, and to Einstein's dismay, he realized that for the theory to be complete it had to take on some absolute characteristics. That's where treating spacetime as the reference for non-uniform motion comes in. It's not just vacuum, according to Einstein (though he pained him to take this view) it's SOMETHING, not just a handy way of expressing seperation. He was forced to move away from the relationist point of view (actually, Einstein and those working with him disliked the name "theory of relativity" because it ignored the important absolute nature of spacetime. It can be curved due to the mass of objects, something that wouldn't happen if it was simply a useful idea. According to Einstein, even in an empty Universe the water would still take on the concave surface- something Mach vehemently denied. So, the problem is your trying to answer an Einsteinian problem with Machian reasoning. Before dealing the principle of relativity and the equivalence principle, you have to first accept that MP's centre of mass reference point is incorrect, or at the very least not applicable, as Einstein didn't follow this reasoning when formulating his theory, or else it's pointless to try solve the problem. Yes, you're correct in saying that the equivalence principle only makes sense in the presence of gravitating masses. There's no point in equating acceleration with gravity if there is no gravity. And it does tie strongly with the principle of relativity. However, abondoning the no longer relevant Machian viewpoint, the principle of relativity applies just as well in the presence of matter as it does not in the presence of matter. Centres of masses are no longer preferred viewpoints. Even in the presence of a gravitating body, all viewpoints are still correct. Of course I may have completely missed your point, if so I apologize.

 

[Haelfix]

'yet the photon has lost energy even though it has traveled across curved space-time on a null-geodesic instantaneously with no forces acting on it. '

That it has traveled in a curved spacetime is precisely why it has lost energy and been redshifted.. The Earth's field gains energy, and the photon loses it. This is a measurable effect. This experiment incidently was performed by Pounds et al in the 60s and verified to 1% accuracy. I think Misner talks about this experiment in detail in his book

Btw, GR is not entirely incompatible with Machs principle.. For instance, in a metric with maximal symmetry (killing vectors), and easily defined boundary conditions, Mach's principle is intact (see Thorne, Misner et al). The big crunch solution in the case of FRW is an example of this.

 

[VantagePoint72]

Point taken, I was simply generalizing as it's incombatible as far as the principle of relativity in the given situation goes. Mach's principle may remain intact with maximal symmetry and easily defined boundary conditions, but that's not relevant to my previous post.

 

[Garth]

I fully understand the the reasoning behind the EEP and its consequences, such as photons loosing energy for no apparent reason, the point is I am questionning it! The pound experiment confirms the fact that a photon is red shifted when compared to the atoms it has been emitted from and absorbed by. It does not resolve the question of whether it has been the photon that has lost energy or the apparatus that has gained it. That is a matter of interpretation. Interpreting it my way then violates the EEP, which is precisely what I intend to do and the point of my other questions. Whereas the EEP has been tested to one part in 10^-14, Self Creation Cosmology (SCC) only violates it to one part in 10^-17, three orders of magnitude smaller than present experimental limits. See "Self Creation Cosmology - An Alternative Gravitational Theory"
http://arxiv.org/abs/gr-qc/0405094 to be published in "Progress in General Relativity and Quantum Cosmology Research." Nova Science Publishers, Inc.

 

[VantagePoint72]

Then you're well beyond the extent of my knowledge and capabilities of my reasoning. I hope to hear more about scientific interest in this theory in the future!