The Equivalence Principle...

[coldcreation]

Originally Posted by Boerseun

Is it making sense?

I'm sure that the two fields are indistinguishable. There will be a difference in acceleration when measured from the ceiling and floor of both rooms (earth-based and rocket based). The inverse square law will be operational in both places, and that the difference in time kept (with a device used that measures the frequency at which atoms resonate: an atomic hydrogen-maser clock, a caesium-133 clock, or the likes) will depend on the altitude (floor-ceiling) or direction of motion (floor-ceiling), respectively.

The latter may seem non-intuitive, because one would expect the floor and ceiling (of the rocket-bound room) to be accelerating at exactly the same rate. But, in fact, the clocks would tell you that is not the case, when compared to clocks on the floor.

So, the zero-dimensional point particles will not be required to determine the outcome of the thought experiment.

Relativity is the rule...

An interesting point arises (no pun intended): On earth, all lines of force point towards the center of the earth (or center of gravity?). The question is, would the same effect be operational inside the rocket, i.e., would all lines of force (pseudo-force) tend towards a point equidistant to that of the earths center (viz the 1g).

If not, that could be a way of telling the difference between the two experiments. But in this case, even if a difference was observed, in a test designed to determine where the center of gravity was located, or the vanishing points would tend from the rocket, I'm not sure it would render untenable EEP.

Edit: I would think the lines of force extending from the rocket would converge at infinity, as opposed to 6,357 km (distance to the center of he earth). So that would be a way to tell the fields apart. I'm sure this would NOT render untenable EEP.

Stop making sense, making sense


CC

[modest]

Originally Posted by ronthepon

Or, we could say that ideally the height of the room (in the direction paralell to the force) should tend to zero.

Originally Posted by Boerseun

So the only way in which gravitational attraction and acceleration can be indistinguishable will be for zero-dimensional point particles which won't experience differences in gravitational pull because their tops and bottoms won't experience the inverse square change in gravitational pull - because they don't have tops and bottoms.

Yeah, as the size approaches zero—that's my understanding. The equivalence principle is a very good approximation at sufficiently small scales and approaches being exactly correct as the area approaches zero.

I'm looking for a source....

All experiments will give the same results in a local frame of reference in free fall and in a local frame of reference far removed from gravitational influences.

That is, there is no experiment we can perform that will tell us whether we are in a free falling reference frame (like the elevator above) or in a reference frame far away in space. The consequences of this profound hypothesis are quite remarkable.

(By the way, note the use of the word local. Local means in this context a region of space that is sufficiently small so that the gravitational field can be considered uniform. If the region is too large we would notice the effects of the non-uniform gravitational field. Then the equivalence principle would not apply. Think about what would happen to two particles in a large falling elevator: the particles are falling towards the center of the earth along radial trajectories; therefore, we would see the particles come progressively closer as the elevator approaches the center of the earth. This would tell us that we were near a massive object, like the earth, rather than at a place far away in space. The equivalence principle would not be valid.)

General Relativity

Craig recently demonstrated toward the middle of this post.

The physics of general relativity is not the same as special relativity near a massive body except for a point-sized free-falling frame of reference.

Originally Posted by coldcreation

The inverse square law will be operational in both places, and that the difference in time kept (with a device used that measures the frequency at which atoms resonate: an atomic hydrogen-maser clock, a caesium-133 clock, or the likes) will depend on the altitude (floor-ceiling) or direction of motion (floor-ceiling), respectively.

The latter may seem non-intuitive, because one would expect the floor and ceiling (of the rocket-bound room) to be accelerating at exactly the same rate. But, in fact, the clocks would tell you that is not the case, when compared to clocks on the floor.

I agree that both the rocket and the room are time dilated so that clocks on the ceiling (or nose) tick faster than clocks on the floor (or the aft of the rocket). But, it isn't the difference in gravitational force between the ceiling and floor that causes the time dilation. Gravitational time dilation is proportional to gravitational potential, not gravitational force. In a rigid rocket there is no difference in force between the nose and the aft, but there is a difference in gravitational (or pseudo-gravitational) potential. It's a uniform field in the rocket so that the force doesn't change throughout the ship, but the potential does.

On earth, it isn't a uniform field. Time dilation would then be a very good approximation between a rocket accelerating at some g-force and a room on the surface of a body experiencing some g-force. But, the longer the rocket and the taller the room the more the time dilation between the two situations would diverge because one is a uniform field while the other is not.

That, in any case, is my understanding.

~modest

[Qfwfq]

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. Typically they haven't read the real thing, only some imprecise divulgative stuff.
http://www.alberteinstein.info/galle..._pp146-200.pdf

Originally Posted by Boerseun

Are you in an accelerating rocket, rocketing off at 1g, or are you in a room on earth, experiencing earth's gravitational pull?

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

According to the EP you can't tell whether you are in the accelerating rocket, or in a perfectly uniform gravitational field. See page 150 in that .pdf file, he doesn't say it's exactly the same as being in an Earthbound room. The way he puts it, he doesn't even need to specifically say "perfectly uniform" because it's quite obvious and he wasn't even the first to notice the fact at all.

For a field that isn't perfectly uniform you would have to get further into the matter, differential geometry and general covariance. It is a statement about the first and second order terms in the description of dynamics, for a given point, This is not quite as simple as the rocket example but one can boil it down to saying that, although you need a slightly different rocket for each point of the room, the discrepancies around that point (between the rocket and the room) are infinitesimals of greater than second order.

[freeztar]

I'm not too fond of the phrase "perfectly uniform gravitational field". How can it be "perfectly uniform" if there is a gradient, however infinitesimal?

If we need a different rocket for every change in z inside the elevator, then how does that relate on an atomic level, per Boerseun's question about the atom?

[modest]

Originally Posted by Qfwfq

http://www.alberteinstein.info/galle..._pp146-200.pdf

According to the EP you can't tell whether you are in the accelerating rocket, or in a perfectly uniform gravitational field. See page 150 in that .pdf file, he doesn't say it's exactly the same as being in an Earthbound room.

Nice. Page 154 says it quite succinctly,

For infinitely small four-dimensional regions the theory of relativity in the restricted sense is appropriate, if the co-ordinates are suitably chosen.

For this purpose we must choose the acceleration of the infinitely small (“local”) system of co-ordinates so that no gravitational field occurs; this is possible for an infinitely small region.

In the derivation of GR the equivalence principle applies to an infinitely small region.

~modest

[Boerseun]

Originally Posted by Qfwfq

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. Typically they haven't read the real thing, only some imprecise divulgative stuff.

Not necessarily.

And I'm not claiming to be the first to think about this, either. I just read a book of scientific essays and speeches by Arthur Clarke (the SciFi guy) where he pointed to this particular issue, and how you will not be able to tell the difference when in a uniform gravitational field when such fields don't exist, and can only be said to be in existence for zero-dimensional points.

Originally Posted by Modest

In the derivation of GR the equivalence principle applies to an infinitely small region.

That's about what I was getting at.

And then it could be said to be effective for n-dimensional bodies, too - because any n-dimensional space would be the sum (an infinite sum, to be sure) of all the 0-dimensional points it contains, although a massive amount of integration will be required if you want the exact, precise answer (which you'll never get, 'cause there's an infinite number of points between A and B)

[Qfwfq]

Actually I read quite a lot of stuff by Clarke, Azimov and the likes when I was young but I don't consider it the same as the proper treatments.

Originally Posted by modest

In the derivation of GR the equivalence principle applies to an infinitely small region.

Yes, the phrase "infinitely small region" is a quick and dirty expression used in most textbooks before getting into the actual calculus. A precisely meaningful statement of it is in terms of orders of infinitesimals.

Originally Posted by freeztar

I'm not too fond of the phrase "perfectly uniform gravitational field". How can it be "perfectly uniform" if there is a gradient, however infinitesimal?

Gradient of field or of potential? Infinitesimal doesn't mean zero and even the gradient of field isn't required to be infinitesimal at all so, if there is a gradient of field it isn't a uniform field. Even if it's an infinitesimal gradient of field, it isn't a perfectly uniform field.

Originally Posted by freeztar

If we need a different rocket for every change in z inside the elevator, then how does that relate on an atomic level, per Boerseun's question about the atom?

Even a shift in the horizontal requires a slightly different rocket, although very very slight. An atom is very small, this doesn't make it infinitely small. Differences for small enough displacements are negligible for a given purpose, how small just depends on the field in question and on what's negligible for a given purpose. On the surface of this planet, differences around the room are negligible to most purposes, at the scale of the Bohr radius they are even tinier. The EP remains even in cases of much higher gradients.

[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 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 , 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]

Originally Posted by Boerseun

... 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