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]

Originally Posted by Boerseun

...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]

Originally Posted by Boerseun

...

...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]

Originally Posted by coldcreation

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]

Originally Posted by 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?

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?