Gravitational field versus acceleration

[Jaysal]

Would it be possible for a person in a lift to know if he is in a gravitational field if he measures the gravitational acceleration at the bottom of the lift versus the top as they would differ since gravitational field strength would be different between the bottom and top.
In an accelerating lift this would not be the case so the person inside would know if the lift is accelerating or not.

 

[Ibix]

Yes. The equivalence principle is only true locally. Strictly, it only applies at a point, but there is always finite precision to your measurements so it is always "good enough" over a small region. If you make precise enough measurements, your lift may not count as a small region and tidal effects such as the one you suggest will be detectable.

 

[stevendaryl]

Would it be possible for a person in a lift to know if he is in a gravitational field if he measures the gravitational acceleration at the bottom of the lift versus the top as they would differ since gravitational field strength would be different between the bottom and top.
In an accelerating lift this would not be the case so the person inside would know if the lift is accelerating or not.

Actually, in an accelerating lift, the gravitational acceleration is higher at the bottom of the elevator than at the top. But you're right, the precise way that gravity changes with height is different in the two cases.

The equivalence principle really should be understood as saying that gravity and acceleration affect physics in the same way. The details of how the "gravitational field" varies from place to place can tell you whether it is due to a gravitating mass or not.

 

[stevendaryl]

Actually, in an accelerating lift, the gravitational acceleration is higher at the bottom of the elevator than at the top.

This is kind of surprising, at first, but heuristically you can understand it this way: As the elevator accelerates, its velocity goes up, and it undergoes length contraction. Which means that the distance between the rear of the elevator and the front of the elevator decreases. Which means that the rear of the elevator travels farther than the front. Which means that the rear is accelerating more.

 

[PeroK]

Would it be possible for a person in a lift to know if he is in a gravitational field if he measures the gravitational acceleration at the bottom of the lift versus the top as they would differ since gravitational field strength would be different between the bottom and top.
In an accelerating lift this would not be the case so the person inside would know if the lift is accelerating or not.

The gravitational field may not be due to a spherical body. In general, if the source of the gravitational field does not produce a totally uniform field, then that can be detected experimentally.

However, in theory, you could have a uniform gravitational field inside the elevator.

 

[sweet springs]

Actually, in an accelerating lift, the gravitational acceleration is higher at the bottom of the elevator than at the top. But you're right, the precise way that gravity changes with height is different in the two cases.

So I may say that in both acceleration in IFR and pulled by gravitational field, the lower in altitude, the higher of force. Detailed quantity investigation is necessary to distinguish them.

Some other features to distinguish the two include that
- Acceleration in IFR has flat event horizon in low place that gravitation field does not always have it. A BH has event horizon but of sphere or spheroid shape.
- Acceleration in IFR undertakes no change in horizontal displacement, gravitational field changes because the direction to the center of mass changes by "horizontal" or perpendicular direction to the center of mass.

 

[stevendaryl]

So I may say that in both acceleration in IFR and pulled by gravitational field, the lower in altitude, the higher of force. Detailed quantity investigation is necessary to distinguish them.

Some other features to distinguish the two include that
- Acceleration in IFR has flat event horizon in low place that gravitation field does not always have it. A BH has event horizon but of sphere or spheroid shape.
- Acceleration in IFR undertakes no change in horizontal displacement, gravitational field changes because the direction to the center of mass changes by "horizontal" or perpendicular direction to the center of mass.

Yes, those are differences between Schwarzschild spacetime and flat spacetime in accelerated coordinates. However, the specific differences having to do with spherical symmetry can be made arbitrarily small. The acceleration felt by an observer hovering at constant Schwarzschild radius $r$ is approximately:

$g = \frac{GM}{r^2}$

If we take the limit as $r \rightarrow \infty$ and $M \rightarrow \infty$ while keeping $g$ constant, the effects of spherical symmetry will become negligible. The horizontal displacement of nearby falling objects will become negligible.

 

[PeterDonis]

Moderator's note: An entire subthread that started with a false claim made by a now-thread-banned poster has been deleted.

 

[Jaysal]

The gravitational field may not be due to a spherical body. In general, if the source of the gravitational field does not produce a totally uniform field, then that can be detected experimentally.

However, in theory, you could have a uniform gravitational field inside the elevator.

Would it be possible for a person in a lift to know if he is in a gravitational field if he measures the gravitational acceleration at the bottom of the lift versus the top as they would differ since gravitational field strength would be different between the bottom and top.
In an accelerating lift this would not be the case so the person inside would know if the lift is accelerating or not.

How about looking at it this way... the accelerating lift will require a Force on one end that end will accelerate first and due to the speed of light limit(and speed of cause and effect), the Force applied at the bottom of the lift will take time to reach the top and therefore creating an infinitesimal difference in acceleration.

 

[FactChecker]

How about looking at it this way... the accelerating lift will require a Force on one end that end will accelerate first and due to the speed of light limit(and speed of cause and effect), the Force applied at the bottom of the lift will take time to reach the top and therefore creating an infinitesimal difference in acceleration.

I have a couple of rather naive questions. With a steady force over a long time, would the acceleration have become equal at bottom and top? Wouldn't it be the velocity that remains different and therefore causes a shortening as the speed increases?

 

[Jaysal]

I have a couple of rather naive questions. With a steady force over a long time, would the acceleration have become equal at bottom and top? Wouldn't it be the velocity that remains different and therefore causes a shortening as the speed increases?

The force maby steady but it still takes time to propagate thru he material. If we have a 1 light year long rod and apply force on one end it'll still take 1 year for the force to reach the other end. Yes velocity will be different and also the rate of change of velocity.

 

[Ibix]

Yes velocity will be different and also the rate of change of velocity.

Not for the reason you describe, however. If it were that simple, then if the front of the rod started accelerating at 1g then the back of the rod would also accelerate at 1g - it just wouldn't start for a while. This would lead to a steady lengthening of the rod due to the velocity difference.

The effect you are describing is real, but it's a transient during startup. The rod/lift initially lengthens due to its elasticity (or snaps if it can't handle the forces). But in the steady state its length (as measured by rulers attached to the lift) must be constant. And its length must be decreasing according to rulers at rest in any inertial frame where its velocity is increasing. Thus the acceleration of the back must be higher than that of the front.

 

[Jaysal]

Not for the reason you describe, however. If it were that simple, then if the front of the rod started accelerating at 1g then the back of the rod would also accelerate at 1g - it just wouldn't start for a while. This would lead to a steady lengthening of the rod due to the velocity difference.

The effect you are describing is real, but it's a transient during startup. The rod/lift initially lengthens due to its elasticity (or snaps if it can't handle the forces). But in the steady state its length (as measured by rulers attached to the lift) must be constant. And its length must be decreasing according to rulers at rest in any inertial frame where its velocity is increasing. Thus the acceleration of the back must be higher than that of the front.

Yes agreed once the force is sent to the destination it will continually be acting on the further side.

 

[PeroK]

How about looking at it this way... the accelerating lift will require a Force on one end that end will accelerate first and due to the speed of light limit(and speed of cause and effect), the Force applied at the bottom of the lift will take time to reach the top and therefore creating an infinitesimal difference in acceleration.

The force could be equally distributed across the lift.

Anyway, this sort of engineering consideration is not the point of the equivalence principle. It's like saying in a lift you might hear the motors humming and the cables rattling! That's missing the point entirely.

 

[FactChecker]

If it were that simple, then if the front of the rod started accelerating at 1g then the back of the rod would also accelerate at 1g - it just wouldn't start for a while. This would lead to a steady lengthening of the rod due to the velocity difference.

Good point. So the SR length contraction can not be thought of as a direct consequence of any force and time delay. It is just a mathematical consequence of the constant velocity of light at different relative speeds. Otherwise, the results would change, depending on where the force was applied (front, back, both equally, etc.)

(sigh) That misconception was comforting but now I will have to give it up.

I guess I can imagine that if an accelerating reference frame thinks that it is accelerating simultaneously at all points (regardless of why), then a stationary reference frame would think that the rear started accelerating early and the length would contract due to non-simultaneous acceleration. So the length contraction during acceleration would be consistent with the relativity of simultaneity.

 

[stevendaryl]

How about looking at it this way... the accelerating lift will require a Force on one end that end will accelerate first and due to the speed of light limit(and speed of cause and effect), the Force applied at the bottom of the lift will take time to reach the top and therefore creating an infinitesimal difference in acceleration.

That's true, but it doesn't explain the difference between the acceleration of the top and bottom of an accelerating rocket. Immediately after accelerating starts, the shape of a rocket or elevator is distorted, and the particular type of distortion depends on how you accelerate it. Think of accelerating a spring. If you accelerate it by pulling on one end, the spring will stretch. If you accelerate it by pushing on the other end, the spring will contract.

But springs (and solid bodies in general) tend to spring back to their equilibrium shape. If you pull on one end of a spring, initially the rear will have a smaller acceleration, but as the spring becomes stretched, the acceleration of the rear will increase (because of the spring force). After a while, the spring will reach an equilibrium length that will be independent of whether you accelerated it by pushing one end or by pulling the other end.

According to Newtonian physics, when this equilibrium is reached, the front and rear will experience identical accelerations. But according to Special Relativity, at equilibrium the rear will be accelerating slightly more than the front.

 

[jbriggs444]

After a while, the spring will reach an equilibrium length that will be independent of whether you accelerated it by pushing one end or by pulling the other end.

This sounds wrong. Surely a spring hanging from an elevator ceiling (in tension) will have a longer equilibrium length than one standing on its end on the elevator floor (in compression).

In the Newtonian model, the equilibrium accelerations of the top and bottom of a given spring will be equal, agreed. But the equilibrium lengths of two otherwise identical springs, one in tension and one in compression will not be equal.

 

[stevendaryl]

This sounds wrong. Surely a spring hanging from an elevator ceiling (in tension) will have a longer equilibrium length than one standing on its end on the elevator floor (in compression).

The equilibrium acceleration of the top and bottom of a given spring will be equal, agreed. But the equilibrium lengths of two otherwise identical springs, one in tension and one in compression will not be equal.

Yes, you're absolutely right. The equilibrium length will be determined by the acceleration. The point is that the equilibrium length is not determined by the lag time for signals to travel along the spring (although the transient length may be).

 

[David Lewis]

...the rear of the elevator travels farther than the front. Which means that the rear is accelerating more.

In a different thread, I believe you said not only is the acceleration greater for the bottom of the elevator than the top, but the top and bottom accelerations are also out of synch because of relativity of simultaneity in the elevator frame.

 

[stevendaryl]

In a different thread, I believe you said not only is the acceleration greater for the bottom of the elevator than the top, but the top and bottom accelerations are also out of synch because of relativity of simultaneity in the elevator frame.

I don't remember saying exactly that, but what I think I have said before is that "gravitational" time-dilation on the rocket---the fact that a clock in the front of the rocket seem to be ahead of a clock in the rear of the rocket--can be attributed to a combination of two factors:

1. Relativity of simultaneity
2. Length contraction

As measured in the initial launch frame, there will be a discrepancy between the times shown on the two clocks because the rear clock is traveling slightly faster. But also, if we transform to the instantaneous rest frame of the rocket after it has begun moving, there will be an additional discrepancy due to relativity of simultaneity---even if the two clocks show the same time, according to the "launch" frame, they will show different times in the instantaneous rest frame. Soon after launching, relativity of simultaneity dominates. Long after launching, length contraction dominates. The two effects add together to produce a consistent difference of measured rates of the two clocks.

 

[David Lewis]

The accelerating lift will require a Force on one end. That end will accelerate first and, due to the speed of light limit (and speed of cause and effect), the Force applied at the bottom of the lift will take time to reach the top and therefore creating an infinitesimal difference in acceleration.

If identical forces are simultaneously* applied to the top and bottom of the elevator then I think tensile stress will build up in the walls even though the length* of the elevator does not change.

*From the initial launch observer's standpoint.