The Equivalence Principle...

[modest]

Here is an interesting way of looking at it:

 

...what I described as an infinitesimal rigid rod. Three such rods meeting orthogonally at a common endpoint whose worldline is [math]\gamma ^ \prime[/math] provide the frame for the Cartesian coordinates of a local Lorentz chart. The possibility of erecting such a frame along any timelike geodesic bears witness to the local (approximate) validity of Euclidean space geometry near each freely falling particle. It is the local Euclidean geometry that decides which objects do and which do not qualify as infinitesimal measuring rods at each place. Its validity evidently is not determined by measurements performed with rods, but flows as a mathematical theorem from the assumption that spacetime is a Riemannian manifold.

 

Thus, the foundations of physical geometry do not rest for Relativity on the

fact

that rigid bodies are freely movable—as Helmholtz argued; nor—as Reichenbach claimed—on a convention to ignore universal forces; but—as Riemann anticipated—on hypotheses that lay at the heart of the scheme of things by which we seek to judge and understand the course of events. Two are the basic hypotheses:...

Relativity and Geometry by Roberto Torretti

 

I see what Qfwfq was talking about saying "You're not the first person to raise this question and many have done so with the conviction they've discovered a major flaw in GR". The quote above makes mention of Helmholtz and Reichenbach. A rejection of Reichenbach's interpretation seems very appropriate:

 

For, as Reichenbach explicitly acknowledged, gravitation is itself a "universal force", coupling to all bodies and affecting them in the same manner (1928, 294-6; 1958, 256-8). Hence the choice recommended by "descriptive simplicity" is merely a stipulation that metrical appliances, regarded as "infinitesimal", be considered as "differentially at rest" in an inertial system (1924, 115; 1969, 147). This is a stipulation that spacetime measurements always take place in regions that are to be considered small Minkowski spacetimes (arenas of gravitation-free physics). By the same token, however, consistency required an admission that "the transition from the special theory to the general one represents merely a renunciation of metrical characteristics" (1924, 115; 1969, 147), or, even more pointedly, that "all the metrical properties of the spacetime continuum are destroyed by gravitational fields" where only topological properties remain (1928, 308; 1958, 268-9). To be sure, these conclusions are supposed to be rendered more palatable in connection with the epistemological reduction of spacetime structures in the causal theory of time.

 

Despite the influence of this argument on the subsequent generation of philosophers of science, Reichenbach's analysis of spacetime measurement treatment is plainly inappropriate, manifesting a fallacious tendency to view the generically curved spacetimes of general relativity as stiched together from little bits of flat Minskowski spacetimes. Besides being mathematically inconsistent, this procedure offers no way of providing a non-metaphorical physical meaning for the fundamental metrical tensor [math]g_{\mu \nu}[/math], the central theoretical concept of general relativity, nor to the series of curvature tensors derivable from it and its associated affine connection. Since these sectional curvatures at a point of spacetime are empirically manifested and the curvature components can be measured, e.g., as the tidal forces of gravity, they can hardly be accounted as due to conventionally adopted "universal forces". Furthermore, the concept of an "infinitesimal rigid rod" in general relativity cannot really be other than the interim stopgap Einstein recognized it to be. For it cannot actually be "rigid" due to these tidal forces; in fact, the concept of a "rigid body" is already forbidden in special relativity as allowing instantaneous causal actions. Secondly, such a rod must indeed be "infinitesimal", i.e., a freely falling body of negligible thickness and of sufficiently short extension, so as to not be stressed by gravitational field inhomogeneities; just how short depending on strength of local curvatures and on measurement error (Torretti (1983), 239).

Early Philosophical Interpretations of General Relativity--Critique of Reichenbachian Metric Conventionalism

 

It seems that just what 'an infinitesimal area approximating special relativity' means is quite an in-depth and interesting subject.

 

~modest

[UncleAl]

Einstein's elevator is the Gedankenexperiment. It is inescapable if space is isotropic. This is observed true in the massless sector (photons) and completely untested in the massed sector (e.g., left- versus right-handed atomic mass distributions),

 

Jahrbuch der Radioaktivität u. Electronik 4 411 (1907)

The Collected Papers of Albert Einstein, Vol. 2 English translation, A. Beck, trans. (Princeton University Press: Princeton, NJ, 1989) p. 252

 

The Equivalence Principle is local - no tidal effects. All local centers of mass vacuum free fall along identical (parallel-displaced) minimum action trajectories. The world line of a body immersed in a gravitational field is independent of composition and structure.

 

Composition has been exquisitely tested in the lab,

 

The University of Washington Eot-Wash Group

 

Earth-moon system falling around the sun (lunar laser ranging and the Nordtvedt effect), and astronomically (pulsar PSR J1903+0327 and solar star binary). All compositions of matter, hydrogen atoms to neutron stars, obey the Equivalence Principle to at least fractional parts-per-trillion difference/average.

 

Test mass composition is locked - but no theory of gravitation contains composition. All gravitation theories are geometries. The proper test of spacetime geometry is mass distribution geometry.

 

PURSUING THE LIMITS OF FAILED SYMMETRY

Do left and right shoes violate the Equivalence Principle?

 

The EP comes in at least three flavors: weak, Einstein, and strong,

 

Equivalence principle - Wikipedia, the free encyclopedia

Eötvös experiment - Wikipedia, the free encyclopedia

Calorimetric Equivalence Principle Test

[coldcreation]

... According to the Equivalence Principle, there is no way to tell the difference...

 

There is at least one way for observers to tell the difference between the two situations (even without windows) from inside an accelerating room (in space) and one stationary room located on earth.

 

All one has to do is extend plumb lines (laser plumb lines perhaps) from the ceiling and measure with a very accurate device the distance between the plumb at the floor.

 

The distances between plumbs at the floor would be shorter in the earth bound room, since the lines of force tend toward the center of the earth (or the center of the earth's gravity field), while in the accelerating room in space, the lines of force are parallel (the distance measured between the lines is identical to the distance between them at the ceiling), since there is no center of gravity.

 

This though, is just an artifact of the experiment, and in no way renders untenable EP; there is still an equivalence of gravitational and inertial mass, and the gravitational 'force' experienced locally is the same as the pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference.

 

 

CC

[Qfwfq]

I see what Qfwfq was talking about saying "You're not the first person to raise this question and many have done so with the conviction they've discovered a major flaw in GR". The quote above makes mention of Helmholtz and Reichenbach. A rejection of Reichenbach's interpretation seems very appropriate:

Actually I hadn't meant to go quite that far, I was thinking about a variety of folks that simply hadn't understood the actual formulation of GR and yet thought they could judge it critically. I do, however, get a vague impression that Torretti did not quite understand the work of Reichenbach, from what little I read in those sources.

 

It seems that just what 'an infinitesimal area approximating special relativity' means is quite an in-depth and interesting subject.

It isn't even necessary once things are formulated in differential terms.

 

The Equivalence Principle is local - no tidal effects. All local centers of mass vacuum free fall along identical (parallel-displaced) minimum action trajectories. The world line of a body immersed in a gravitational field is independent of composition and structure.

Eh? The Equivalence Principle is local - yes! But....

 

Geodesic deviation isn't even all that extremely complicated to calculate in the GR formalism. See e. g. the section on it in Weinberg's classic textbook, it's pretty straighfoward. Where do you get the idea that the EP's local formulation implies no tidal effects and geodesics being parallel-displaced?

[UncleAl]

Einstein's elevator Gedankenexperiment demands no divergence, curl, or multipolar contributions to the acceleration field. It also demands isotropic vacuum. The weak Equivalence Principle is exactly as I stated it. Minkowski space is then locally obtained in vacuum free fall.

[Qfwfq]

As so often, here too, you are being quite inconsequential and misleading, apart from not even clearly defining what you say.

 

Einstein's demands no divergence, curl, or multipolar contributions to the acceleration field. It also demands isotropic vacuum.

The elevator Gedankenexperiment demands a uniform field, exactly the first thing I had said in this thread. But that is where the principle applies not only locally. In the Schwartzschild solution, if you consider a [imath]\Delta r[/imath] displacement and parallel transport the tangent vector at one point of a timelike geodesic, the resulting geodesic won't all be the parallel displacement of the first one. For a displacement having [imath]\Delta r=0[/imath] the difference will just be of higher order in the spatial distance; the accelerations are equal iff the distance is zero. Do you disagree with this? In exactly what sense do you claim "no tidal effects" as a consequence of the principle being local?

 

The weak Equivalence Principle is exactly as I stated it.

Exactly what did you state? Don't hide behind a pane of glass, clarify.

 

Minkowski space is then locally obtained in vacuum free fall.

Red herring. A chart can always be chosen so that for a given point the tangent space is Minkowskian and acceleration of particles at is zero the same point. The word "locally" means this might not be possible throughout a finite region with one same chart; it means that tidal effects are not an objection, it does not mean that there are none. So, what is your point?

 

See the English translation of Die Grundlagen that I linked to and work out the calculus. It is possible in a finite region iff the field is uniform in it; else the principle applies only locally (which means there are tidal effects).

[UncleAl]

Weak Equivalence Principle, Einstein EP, strong EP. You are confusing their particulars. Some folks add the very strong EP. Push come to shove, present two lumps that violate the EP and the whole house of cards collapses. Present two chemically and macroscopically identical lumps that violate the EP and...

 

PURSUING THE LIMITS OF FAILED SYMMETRY

Somebody should look.

[Qfwfq]

Weak Equivalence Principle, Einstein EP, strong EP. You are confusing their particulars. Some folks add the very strong EP.

In GR literature and courses I have come across the EP in the weak, strong and very strong sense. I was not confusing their particulars; they apply in differential form except in absence of gravitating objects unless you consider an indefinitely extended homogeneous slab. For a finite region it is approximated if massive bodies are at huge distance. What is your definition of the "Einstein EP" and how do you distinguish it from the others?

 

Push come to shove, present two lumps that violate the EP and the whole house of cards collapses. Present two chemically and macroscopically identical lumps that violate the EP and...

 

PURSUING THE LIMITS OF FAILED SYMMETRY

Somebody should look.

Are you aware that geometric reflection is a quite distinct matter from spin? The link is via orbital angular momentum, but nowhere in all your ranting about shoes and chirality do you effectively support the idea that there should be a distinction between L and R molecules or crystals in a quasi-uniform gravitational field. Wouldn't moment of inertia about any corresponding c. m. axes be the same, for any two parity opposite objects? Can you afford any other nexus between these and spin (intrinsic angular momentum)? Have you checked if gyroscopes behave otherwise than what Thomas precession predicts?

 

Where some folks want to look is where spin is relevant along with gravity, in the Cartan variant of GR the source of torsion would presumably be spin. Neutron stars could possibly have macroscopic spin so astrophysicists ought to look for evidence around there. The Weitzenböck version is observationally equivalent, formally it is just a different choice of gauge.

 

You confuse various things in your illucid rantings.

[UncleAl]

Backreaction: The Equivalence Principle

[Boerseun]

That is what Einstein explains in his thought experiment with the elevator. If you are standing in the elevator (that is just a local patch, theoretically infinitesimally small)...If you could make your elevator larger you could however eventually distinguish between flat and curved space because you could measure geodesic deviation, i.e. the curvature.

That's exactly what I was getting at. UncleAl, you're a rock star, dude.

[Qfwfq]

Boerseun, hilarious as that page is, what you quote had already been said with much less beating around the bush than he did. I had invited him to clarify what he meant by:

The Equivalence Principle is local - no tidal effects.

which isn't at all equivalent to your quote from that site. That (and the remainder of what I had quoted then) simply don't match up, to a general relativist. Let's forget about what he followed with.

 

Unk, you could well have linked and quoted those words much sooner, if you felt it hadn't been said too clearly. I know Boerseun is not so specialized in differential calculus, but given your carry-on I'm not sure you were any more able than him to recognize that fact in what had been said. Otherwise I ought to suppose you simply meant to clarify it for his greater ease. If so Unk, you could have clarified whopping mighty sooner.B)

[Boerseun]

My question in the OP:

So does it hold only for imaginary point-particles?

...with the entire explanation about gravity being propagated at an inverse square, thus giving the game away in the room, making the difference between a space-bound room and an earthbound room detectable.

UncleAl's statement that:

The Equivalence Principle is local - no tidal effects.

says the same, albeit précis-wise, as

That is what Einstein explains in his thought experiment with the elevator. If you are standing in the elevator (that is just a local patch, theoretically infinitesimally small)...If you could make your elevator larger you could however eventually distinguish between flat and curved space because you could measure geodesic deviation, i.e. the curvature.

I will grant you that the page was written tongue-in-cheek, but I would not really call it hilarious. It's not really all that funny. But it does kinda answer my question in the OP.

 

The Equivalence Principle assumes a perfectly uniform gravitational field for the earthbound room. A perfectly uniform gravity field does not exist, and can only be said to be experienced by points with no dimension in the direction of the gravitational pull, lest tidal effects give the game away. Thus UncleAl's statement about the tidal effects.

 

I may not be specialized in lots of things, amongst others English. I have studied this particular matter, but in Afrikaans. My translations might be lacking at times - for that I apologize. My statement about

Because the earth-bound room will return two different readings for gravity at the floor and ceiling, because gravity is propagated at an inverse square.

can simply be stated as "tidal effects", as per UncleAl.

 

I find his postings and links in this thread entirely satisfactory regarding the OP, and I think its neither appropriate nor desirable to drag in any of UncleAl's prior transgressions (be it real or not) with sentences like

As so often, here too, you are being quite inconsequential and misleading, apart from not even clearly defining what you say

or even

Unk, you could well have linked and quoted those words much sooner, if you felt it hadn't been said too clearly. I know Boerseun is not so specialized in differential calculus, but given your carry-on I'm not sure you were any more able than him to recognize that fact in what had been said. Otherwise I ought to suppose you simply meant to clarify it for his greater ease. If so Unk, you could have clarified whopping mighty sooner.

I find the last sentence to be a rather patronizing and, quite truthfully, ridiculous. My personal opinion. I do not understand your attitude regarding UncleAl in this thread. But I will remind you that threads evolve, threads go left and right, up and down, and not according to any particular member's whims or wishes. You cannot realistically expect UncleAl (or any other member) to make a statement or clarification at the exact right time you expect him or her to. It does not work that way.

 

Mighty sorry and brim-full of apologies if this post comes across as inappropriate or rude, but if you have any issues with UncleAl, please take it off-line. PM him, or something. I found his posts here perfectly suitable and applicable to the OP.

[Qfwfq]

If you are satisfied with his answer, and weren't satisfied with some mighty equivalent ones much earlier, even less with some more precise ones and if you even like being mislead by his specious statements, that's your choice but it still stands that he brought his fallacious arguments in here as well and I find no reason for me not to contradict them.

 

Mighty sorry if you thought I was criticizing you in this thread, but I wasn't. Rather, I thought your problem was somewhat caused by too many folks (including some authors of divulgation) making it seem as if

The Equivalence Principle assumes a perfectly uniform gravitational field for the earthbound room.

which is simply not the case, if you're still saying that it appears you still haven't quite got the point of what GR really says and what it doesn't.

 

I think your English is very good and I don't think this is the problem at all and I did not criticize that page, I found it humourous; you missed my poiint. If you aren't deep into differential calculus and the subtleties of infinitesimals, you can be content with more intuitive explanations but you must expect there could be things you're prone to miss; this doesn't exactly enable you to judge every statement in GR as correct or fallacious.

 

Thus UncleAl's statement about the tidal effects.

Actually his statement could be more easily taken the opposite way. Of course there are no tidal effects if you consider a point and not its neighbourhoods; this is not what GR is all about. GR discusses neighborhoods (which is very distinct from discussing the single point) and guess what, where the field gradient isn't zero there is a tidal effect even in an infinitesimal neighborhood.

 

He literally said "no tidal effects" which doesn't contribute to making this forum informative. Only somebody who already understands the matter can suspect he just might be trying to say that tidal effects don't contradict the principle as expressed in local form (which BTW means it does not assume uniform field at all).

 

But I will remind you that threads evolve, threads go left and right, up and down, and not according to any particular member's whims or wishes. You cannot realistically expect UncleAl (or any other member) to make a statement or clarification at the exact right time you expect him or her to. It does not work that way.

This is not a reason to tell me what to do and not. What I said to him was perfectly justified and it wasn't a matter of "at the exact right time I expect".

[CraigD]

I think this thread has managed to answer the original question, which I paraphrase

does General Relativity state that it’s impossible to distinguish an isolated room subjected to constant acceleration from one subject to gravitational acceleration?

After trying both possible answers – yes and no – I believe we’ve shown the answer is

no, for any room with non-zero height, it’s possible to distinguish forces due to gravity from forces due to the room’s acceleration.

There are many ways to state this, some involving the terms “equivalence principle” “infinitesimal” and “tidal force”, but all describe, I think, the same underlying principle.

 

Simple as this is, I’ve long had a problem applying it. In particular, describing what an un-accelerated observer would observe an accurate clock on a body under acceleration to read relative to his own accurate clock, where both have small masses, and are far from large-mass bodies.

 

Consider two spaceships far from any star or planet. Initially, A has a velocity of 1000 m/s nearly directly toward B. The distance between A and B is 5000 m. A has an acceleration of 100 m/s/s (a bit over 10 gs). After 10 seconds, A and B are near one another, with relative velocity 0.

 

Around that instant, what rate relative to his accurate clock would an observer on B observe of an accurate clock on A?

 

The formula for gravitational time dilation in an accelerated box,

[math]T_d = e^{a h /c^2}[/math]

, appears at many reference websites, including this wikipedia page. [math]a[/math] is acceleration, but [math]h[/math] is “is the ‘vertical’ distance between the observers”, which I’m unable to understand.

 

Does anyone know how to calculate the answer to the question above?

[Qfwfq]

Simple as this is, I’ve long had a problem applying it. In particular, describing what an un-accelerated observer would observe an accurate clock on a body under acceleration to read relative to his own accurate clock, where both have small masses, and are far from large-mass bodies.

 

Consider two spaceships far from any star or planet. Initially, A has a velocity of 1000 m/s nearly directly toward B. The distance between A and B is 5000 m. A has an acceleration of 100 m/s/s (a bit over 10 gs). After 10 seconds, A and B are near one another, with relative velocity 0.

 

Around that instant, what rate relative to his accurate clock would an observer on B observe of an accurate clock on A?

Strictly, this is not a problem of GR, it is SR. At that instant, [imath]dt'=dt[/imath]. From the inertial rest frame, the ratio [imath]\frac{dt'}{dt}[/imath] is just according to the usual formula with the instantaneous value of [imath]v^2[/imath], you can even calculate for a finite interval by integrating the same formula with the time dependance [imath]v^2(t)[/imath].

 

The formula for gravitational time dilation in an accelerated box,

[math]T_d = e^{a h /c^2}[/math]

, appears at many reference websites, including this wikipedia page. [math]a[/math] is acceleration, but [math]h[/math] is “is the ‘vertical’ distance between the observers”, which I’m unable to understand.

Gee that section is rather poorly worded. Here, the observers are mutually at rest so both have the same velocity. The [imath]h[/imath] is simply their distance according to the direction of the acceleration and the numerator in the exponent is just the potential difference for a uniform field (and the denominator of course is due if units aren't natural). The exponential simply maps an additive comparison onto a multiplicative one because you want the ratio [imath]\frac{\Delta t'}{\Delta t}[/imath] (no integration is necessary).

[modest]

After 10 seconds, A and B are near one another, with relative velocity 0.

 

Around that instant, what rate relative to his accurate clock would an observer on B observe of an accurate clock on A?

 

At that instant neither are accelerating nor have relative velocity,so t = T. No difference in rate. But, I think you mean, what is the rate during the acceleration...? But the rate changes. Because A is accelerating and B is not accelerating there is not a constant difference in rates. The velocity between the two changes so time dilation comes from acceleration and velocity. The equation:

[math]T_d = e^{a h /c^2}[/math]

only works if the distance between them (h) is always the same. That is, if both A and B were accelerating at the same rate, in the same direction, then the equation above would give the time dilation between them, where h is the distance between them.

 

If you want to find the difference in time for your whole thought experiment, from start to finish, between the two you can use the relativistic rock equation. If the trip takes exactly 10 seconds from A's perspective then B's clock would advance:

[math]t=\sqrt{\left( \frac{d}{c} \right) ^2 + \frac{2d}{a} } = \sqrt{\left( \frac{5000}{299792458} \right) ^2 + \frac{2 \cdot 5000}{100} } = 10.0000000000139 \ s[/math]

 

~modest