The Equivalence Principle...

[Boerseun]

You're stuck in a room with no windows.

 

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.

 

There's a problem with this:

 

If you've got some tools and instruments with you, you can clearly tell the difference, because the gravitational pull should be bigger on the room's floor than on the ceiling. If you're stuck in a rocket, the 1g acceleration will be the same all over the room.

Because gravity is inverse square, and accelerating a body uniformly will make for a uniform experience of 1g, there can be no comparison.

 

The example of the 1g accelerating rocket/room on earth makes sense on an intuitive level because we can easily picture it, but on a fundamental level it does not hold.

 

Am I missing something here? Will spatial contraction in the direction of travel make for a different g-reading at the floor and the ceiling of an accelerating spaceship? If so, it should be measurable to exactly counter the difference in gravity felt at the height of the ceiling compared to the floor for the room on earth, and also at an inverse square rate.

 

If not, the Equivalence Principle will not hold even for atoms, because the top and bottom of atoms will measure gravity differently (minuscule, but still...)

 

So does it hold only for imaginary point-particles?

[ronthepon]

From what I understand of this matter, the principle should not be taken so literally. The 'room' is described only as a simplification, and we should pretty much assume that the variation of g throughout the room is minuscule and outside the measuring capacity of the individual inside.

 

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

 

That's just about the most educated guess I could build up.

[coldcreation]

...Will spatial contraction in the direction of travel make for a different g-reading at the floor and the ceiling of an accelerating spaceship? If so, it should be measurable to exactly counter the difference in gravity felt at the height of the ceiling compared to the floor for the room on earth, and also at an inverse square rate.

 

I love these kind of questions (it's a good one ;)). You got me thinking. I wanted to answer with intuition alone. Could not. So looked it up. Found nothing. And so will try with intuition (or counter-intuition).

 

Intuition could lead to the first conclusion that both fields are slightly different, as you mention. But counter-intuition, whatever that is, leads to the conclusion that both fields are exactly identical. Indeed, I would have to come to the conclusion that the field caused by (or resulting from) the acceleration of the room in the rocket would mimic perfectly the acceleration caused by (or resulting from) the gravitational field of earth (in your example).

 

So there would be a slight difference in the force and pseudo-force between floor and ceiling in both cases respectively, and the difference would be exactly the same. There would, hence, be no way to tell the difference from inside the room (unless there were windows out of which one could look), even with high-precision measuring devices (of, say, the kind use in the Pound-Rebka experiment, and subsequent devices use to test gravitational redshift and/or time dilation).

 

I'll be back to check out what others write on the topic...:(

 

CC

[coldcreation]

I did ended up looking into this question further. and found that the fields must be indistinguishable:

 

The equivalence principle proper was introduced by Albert Einstein in 1907, when he observed that the acceleration of bodies towards the center of the Earth at a rate of 1g (g = 9.81 m/s2 being a standard reference of gravitational acceleration at the Earth's surface) is equivalent to the acceleration of an inertially moving body that would be observed on a rocket in free space being accelerated at a rate of 1g. Einstein stated it thus:

 

"we [...] assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system." (Einstein 1907).

 

That is, being at rest on the surface of the Earth is equivalent to being inside a spaceship (far from any sources of gravity) that is being accelerated by its engines. From this principle, Einstein deduced that free-fall is actually inertial motion. By contrast, in Newtonian mechanics, gravity is assumed to be a force. This force draws objects having mass towards the center of any massive body. At the Earth's surface, the force of gravity is counteracted by the mechanical (physical) resistance of the Earth's surface. So in Newtonian physics, a person at rest on the surface of a (non-rotating) massive object is in an inertial frame of reference. These considerations suggest the following corollary to the equivalence principle, which Einstein formulated precisely in 1911:

 

 

"Whenever an observer detects the local presence of a force that acts on all objects in direct proportion to the inertial mass of each object, that observer is in an accelerated frame of reference."

 

 

Einstein also referred to two reference frames, K and K'. K is a uniform gravitational field, whereas K' has no gravitational field but is uniformly accelerated such that objects in the two frames experience identical forces:

 

"We arrive at a very satisfactory interpretation of this law of experience, if we assume that the systems K and K' are physically exactly equivalent, that is, if we assume that we may just as well regard the system K as being in a space free from gravitational fields, if we then regard K as uniformly accelerated. This assumption of exact physical equivalence makes it impossible for us to speak of the absolute acceleration of the system of reference, just as the usual theory of relativity forbids us to talk of the absolute velocity of a system; and it makes the equal falling of all bodies in a gravitational field seem a matter of course." (Einstein 1911)

 

This observation was the start of a process that culminated in general relativity. Einstein suggested that it should be elevated to the status of a general principle when constructing his theory of relativity:

 

"As long as we restrict ourselves to purely mechanical processes in the realm where Newton's mechanics holds sway, we are certain of the equivalence of the systems K and K'. But this view of ours will not have any deeper significance unless the systems K and K' are equivalent with respect to all physical processes, that is, unless the laws of nature with respect to K are in entire agreement with those with respect to K'. By assuming this to be so, we arrive at a principle which, if it is really true, has great heuristic importance. For by theoretical consideration of processes which take place relatively to a system of reference with uniform acceleration, we obtain information as to the career of processes in a homogeneous gravitational field." (Einstein 1911)

 

In other words, if the fields were different, GR would be invalid.

 

CC

[Boerseun]

Well, the problem I have with K and K' is that if spatial contraction were to cater for the difference in measuring acceleration between the ceiling and floor of the rocket room, then it should also be inverse square if you're measuring in the accelerating rocket, because its supposed to mimic identically the inverse square of gravity as measured in the static gravity-bound room on earth.

 

...but then, spatial contraction would also prevent any instrument from measuring that difference, because that instrument will be subject to the same contraction...

 

Which means that if the Equivalence Principle was to hold, it can only hold for zero-dimensional points which would experience gravity/acceleration uniformly.

 

Er...

 

Hmmmm... :(

[freeztar]

It depends on the frames used. Are we judging the atom as a whole or breaking it up into regions?

[coldcreation]

...

 

...but then, spatial contraction would also prevent any instrument from measuring that difference, because that instrument will be subject to the same contraction...

 

Which means that if the Equivalence Principle was to hold, it can only hold for zero-dimensional points which would experience gravity/acceleration uniformly.

 

Er...

 

Hmmmm... :(

 

Where's modest when you need him? ;)

 

Surely the inverse square law would hold true for both situations, and from wherever you measure in the room.

 

I'm guessing that the same effect of 'contraction' will take place in both experiments. So the result will be identical, regardless of the size of the measuring apparatus.

 

CC

[Boerseun]

Where's modest when you need him? :(

 

I'm guessing that the same effect of 'contraction' will take place in both experiments. So the result will be identical, regardless of the size of the measuring apparatus.

 

CC

I don't quite think so, because the gravitational difference between the ceiling and floor of the earth-bound room is actually measurable.

 

The thing is - the very same instrument should provide the very same difference in floor and ceiling readings in the rocket room - which it can conceivably only do if the floor and ceiling were accelerating at different rates - or if spatial contraction occurred at the exact rate to make up for the inverse square of the gravitational field experienced by the earth room - which won't be measurable by an instrument in the same frame of reference...

 

Heck - now I'm seeing double....

[freeztar]

Length contraction is only significant for "outside" observers.

 

The Earth-box occupants will measure floor to ceiling just the same as the space-box occupants (given both frames experience the same acceleration). :(

 

Perhaps I misunderstand your assertion, B?

[Boerseun]

Length contraction is only significant for "outside" observers.

 

The Earth-box occupants will measure floor to ceiling just the same as the space-box occupants (given both frames experience the same acceleration). :(

 

Perhaps I misunderstand your assertion, B?

I think you might have.

 

What I said is that the two rooms, one on earth and one in an accelerating spaceship, won't be indistinguishable as per the Equivalence Principle.

 

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. In the accelerating rocket room, the reading at the floor and ceiling will be identical. So I was considering ways in which a rocket room will give different floor and ceiling readings, and considered spatial contraction, but also pointed out that it will not be measurable because the instrument will be in the same frame and experience the same contraction so will not be able to indicate any possible change.

 

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.

 

...or something like that ;)

 

Is it making sense?

[coldcreation]

 

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 :Music:

 

 

CC

[modest]

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

 

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.

 

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/gallery/pdf/CP6Doc30_English_pp146-200.pdf

 

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]

http://www.alberteinstein.info/gallery/pdf/CP6Doc30_English_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]

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.

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 ;)

[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.

 

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.

 

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.

 

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.