Equivalence Principle

It makes sense, but I don't think it's right!

The gradient of the gravity field exists for a spherical mass such as a planet, but in principle the field could be constant over a certain region of space. E.g. a rectangular slab of matter. Not likely to exist in nature, admitedly, but OK as a thought-experiment (Einstein went in for them!)

I'm intrigued by "Our current technology can measure a change in gravity with less than a centimeter of elevation change on the surface of the earth". Any idea how this is achieved? I estimate the difference as about 1 part in 4*10¹⁷, which is very impressive if it's possible.

Cheers..........Codey

True that an infinite slab has a zero gravitational gradient, but as for the hypothetical elevator, I think my reasoning proves it false with just a single contrary example. Not at all that I think the equivalency principle is wrong, just that the elevator example isn't true. Some other cool ones are that gravity is zero at the center of the earth, or anywhere inside a hollow sphere.

As for measuring gravity with high precision, there are relative gravimeters that use a mass balanced on a lever arm with a zero length spring (meaning that if the coils could physically collapse upon themselves, the spring would have zero length). These are basically low frequency seismometers, check out a company called Scintrex who makes these springs out of quartz. They're sensitive to a single uGal (a Gal = 1cm/s^2). With a free air gradient of 3.086 uGal/cm, that's about a 3 millimeters of elevation change and you can start measuring the difference in gravity. These types of instruments only give a difference in gravity between points though, hence the name of a relative gravimeter. They're are used quite a bit in geo-exploration fields.

For measuring absolute gravity and getting an actual value at a single location, the most accurate device we have measures a falling mirror in a vacuum with an atomic clock and an interferometer (counting and timing the interference fringes of an iodine stabilized HeNe laser) to get the acceleration. A lot like a cop's radar, but a whole lot more accurate. These are accurate to 1-2 uGals. These are made by a company called Micro-g Lacoste... pretty cool instruments.

Interesting!

BTW I dropped a cod in my calculation. I got 4*10¹⁷ from (Earth radius/1 cm)² but that's irrelevant. Doing it again I get your figure 3.086 uGal/cm.

"Some other cool ones are that gravity is zero at the center of the earth, or anywhere inside a hollow sphere."

The one that I like is that a bullet fired through a hole drilled all the way through the earth would get to the target, on the far side, more quickly than if fired the same distance through open space.

care to elaborate? BBB

You are correct that, in principle, you could detect the difference between an accelerating elevator and the gravitational field of the earth. If you were to compare the accelerating elevator to the gravitational field of a large plane solid, you'd see no difference. No, I don't know where you'll find one of those solids.

Experiments with torsion balances have established the validity of the equivalence principle however.

You are correct, the accelerating elevator is only a very close approximation, but in Einstein's time you couldn't physically distinguish the two, except in theory. Not only does the magnitude vary from top to bottom, but the direction varies from side to side.

The point of the elevator example is not the background gravitation of the earth, but rather the addtional force due to the initial upward acceleration. When it starts going up, it's as if the gravity of the earth increases while accelerating upward, and vice versa for the downward journey. That additional force is what the elevator equivalence principle example is describing.

H albery, you said: "If you have any distance to make a measurement, at the top and bottom of the elevator for example, you'll measure a gradient if you're in a gravitational field, but not if you're simply accelerating."

Not quite true. An accelerometer at the top of a uniformly accelerating elevator measures a smaller acceleration than one on the floor, just like for accelerometers at the top and the base of a tower on a planet.

This is a true, but not a very well-known feature of relativistic acceleration.

-J

I should perhaps have qualified "uniformly accelerating elevator" better. I meant uniform (constant) acceleration over time, not uniform over the height of the elevator. What I described also has nothing to do with mechanical compression/stretching of the elevator - it is a pure relativistic effect.

-J

Interesting Jorrie

What is the magnitude of this effect? (at say 1-g, if it varies with acceleration, as I assume it does). Just to compare with variation of g above Earth, ref Albery's figure in #3.

Cheers...........Codey

Hi Codey

The difference in accelerometer readings for a 10m high 'elevator', accelerating in free space at 1g, is about 3.4 μg. It is roughly the same as the difference between accelerometers sitting at the bottom and the top of a 10m high tower on Earth.

It may be a trifle too small to measure in practice, I think, but I may be overly pessimistic...

-J

Hello Jorrie

Thanks for that. You say "It may be a trifle too small to measure in practice, I think" - that's what I would have thought too, but Albery tells us difference can be detected over 3 mm!! which is very impressive.

I assume it's a coincidence that the difference over elevator height 10m is ~ same as difference in g over 10m tower, as for a given value of g, the gradient of g is higher for a body of smaller radius (and mass).

Cheers.......Codey

Hi Codey, sorry I got units screwed up - the effect is much smaller for 1g. It approaches the Earth gravity differential per meter height only at 100 km length! Tiny indeed.

Nevertheless, the effect is there and will be measurable at large accelerations. It is fully described by H. Nikolic in a paper: "Relativistic contraction of an accelerated rod".

The equation he gives is: a' = a/(1+aL/c²), where L is the length. The differential will then be a - a'.

-J