Help with equivalence principle please.

[rede96]

I am just trying to understanding of the equivalence principle and was hoping someone would be kind enough to sense check my thoughts below.

1) If I was in a rocket ship accelerating at a rate of 1g relative to the earth, the clock on my ship would tick at the same rate as a clock on the earth.

2) If I was to then decelerating at the rate of 1g (i.e. negative acceleration) wrt to the earth, although the force I would feel would be repelling me wrt to the earth, the two are still equivalent. So a clock on my ship would tick at the same rate as a clock on earth.

3) If I was to travel in a circle around the Earth such that the net force I felt was equal to 1g, although the force I would feel would be repelling me wrt to the earth, the two are still equivalent. So a clock on my ship would tick at the same rate as a clock on earth.

4) If the above are true, why is there no time dilation or length contraction, as I am moving with wrt the earth?

Thanks!

 

[JesseM]

The equivalence principle doesn't deal with comparisons of distant clocks (that would require a global simultaneity convention), it only deals with the way the laws of physics look within a tiny patch of spacetime around a freefalling observer where a "local inertial frame" can be defined (see this article for a discussion)

 

[PAllen]

I am just trying to understanding of the equivalence principle and was hoping someone would be kind enough to sense check my thoughts below.

1) If I was in a rocket ship accelerating at a rate of 1g relative to the earth, the clock on my ship would tick at the same rate as a clock on the earth.

2) If I was to then decelerating at the rate of 1g (i.e. negative acceleration) wrt to the earth, although the force I would feel would be repelling me wrt to the earth, the two are still equivalent. So a clock on my ship would tick at the same rate as a clock on earth.

3) If I was to travel in a circle around the Earth such that the net force I felt was equal to 1g, although the force I would feel would be repelling me wrt to the earth, the two are still equivalent. So a clock on my ship would tick at the same rate as a clock on earth.

4) If the above are true, why is there no time dilation or length contraction, as I am moving with wrt the earth?

Thanks!

1) yes and no. Yes it would run slow relative to a nearby stationary clock by the same amount as an Earth clock relative to a clock in free fall away from the earth. However, as it sped up from its acceleration, relative to a clock on earth, it would get slow.
2) yes and no, as above.
3) More complex. You have two effects to consider Earth's curving of spacetime, and your motion. The answer would depend on details of your motion. For example, imagine a circular orbit around the Earth (no g force felt), versus holding steady position above Earth with a rocket (some g force felt). The orbiting clock will actually run slower than the stationary clock. On the other hand, a clock shooting away from the Earth and falling back will go faster the the stationary clock. However, you asked about moving such that 1 g is felt. This is much larger than the Earth's gravity at an orbital height, so this effect should be dominant at first. However, as your speed increased from your 1 g accelaration in a circle, the time dilation from this speed would soon dominate, and your clock would run slower than on earth.
4) What are you asking. Time dilation and length contraction are relative effects. In relation to what?

[(1) and (2) are still not quite right. See post #12 for further discussion.]

 

[JesseM]

1) yes
2) yes

Why do you say that? The equivalence principle says nothing about simultaneity between distant events, and without doing so there's no way to compare the rate a clock on a ship traveling at 1G in space and the rate of a clock on Earth.

 

[PAllen]

Why do you say that? The equivalence principle says nothing about simultaneity between distant events, and without doing so there's no way to compare the rate a clock on a ship traveling at 1G in space and the rate of a clock on Earth.

Yes, I thought of that after a minute, now corrected. For a rocket a million miles from from Earth (for example), it would not be hard to set up simultaneity and a near inertial frame in which the Earth was effectively stationary.

 

[rede96]

The equivalence principle doesn't deal with comparisons of distant clocks (that would require a global simultaneity convention), it only deals with the way the laws of physics look within a tiny patch of spacetime around a freefalling observer where a "local inertial frame" can be defined (see this article for a discussion)

Great link, thanks :)

I was actually looking at this from a slightly different angle. Assume I want to accelerate away from the Earth at a rate of 1g for a known distance (x). I calculate the time (t) it would take me to get there, from the earth’s frame of reference.

I then set off at a constant acceleration of 1g, monitoring the time on my clock. When I reach the known distance (assuming I am still accelerating) would the time on my clock show the calculated time (t)?

I know that if I was traveling in uniform motion at relativistic speeds wrt the earth, then my clock would show less elasped time then calculated.

I also suspected that if I accelerated at a greater rate than 1g that it might also show less elapsed time.

But if my acceleration matched the gravitational force on earth, and was therefore equivalent, would there be no difference to the calculated time?

 

[JesseM]

Yes, I thought of that after a minute, now corrected. For a rocket a million miles from from Earth (for example), it would not be hard to set up simultaneity and a near inertial frame in which the Earth was effectively stationary.

But in your modified version, isn't it irrelevant whether the acceleration is 1G or 0.5G or 10G? All that matters is that the initial velocity is at rest relative to the Earth clock, so then in your near inertial frame the two clocks will tick at about the same rate until a significant velocity difference has built up (edit: but actually, wouldn't there still be gravitational time dilation for the clock on Earth in a "near inertial frame", so they wouldn't tick at the same rate?) So, this doesn't really seem to have anything to do with the equivalence in local observations between being at rest on Earth and accelerating in space at 1G.

 

[PAllen]

Great link, thanks :)

I was actually looking at this from a slightly different angle. Assume I want to accelerate away from the Earth at a rate of 1g for a known distance (x). I calculate the time (t) it would take me to get there, from the earth’s frame of reference.

I then set off at a constant acceleration of 1g, monitoring the time on my clock. When I reach the known distance (assuming I am still accelerating) would the time on my clock show the calculated time (t)?

I know that if I was traveling in uniform motion at relativistic speeds wrt the earth, then my clock would show less elasped time then calculated.

I also suspected that if I accelerated at a greater rate than 1g that it might also show less elapsed time.

But if my acceleration matched the gravitational force on earth, and was therefore equivalent, would there be no difference to the calculated time?

No, it is more complex. Two effects are involved. First, you change in distance from Earth (less gravitational slowdown), secondly your increase speed. You cannot make the second go away by going at 1 g. Further, the Earth's gravity gets weaker as you get farther away, so the 1 g is wrong. The dominant effect would be time dilation from speed if the distance was significant.

 

[rede96]

No, it is more complex. Two effects are involved. First, you change in distance from Earth (less gravitational slowdown), secondly your increase speed. You cannot make the second go away by going at 1 g. Further, the Earth's gravity gets weaker as you get farther away, so the 1 g is wrong. The dominant effect would be time dilation from speed if the distance was significant.

Ah, ok. Thanks. Didn't think of that.

So using the inverse square law, if I accelerate away from the Earth so my net effect of my acceleration matches the pull of the Earth's gravity, can I get both clocks to tick at the same rate?

 

[PAllen]

Ah, ok. Thanks. Didn't think of that.

So using the inverse square law, if I accelerate away from the Earth so my net effect of my acceleration matches the pull of the Earth's gravity, can I get both clocks to tick at the same rate?

You still have two competing effects. However, there will be some trajectory where your clock will remain synchronous with the ground. This follows because there is a trajectory where your clock will be faster (bullet trajectory) and others where it is slower, thus there must exist some path that maintains synchronization. However, it will not simply be accelerating at the ambient gravitational acceleration at all times.

 

[rede96]

You still have two competing effects. However, there will be some trajectory where your clock will remain synchronous with the ground. This follows because there is a trajectory where your clock will be faster (bullet trajectory) and others where it is slower, thus there must exist some path that maintains synchronization. However, it will not simply be accelerating at the ambient gravitational acceleration at all times.

Does that mean when accelerating, direction affects time dilation?

 

[PAllen]

But in your modified version, isn't it irrelevant whether the acceleration is 1G or 0.5G or 10G? All that matters is that the initial velocity is at rest relative to the Earth clock, so then in your near inertial frame the two clocks will tick at about the same rate until a significant velocity difference has built up (edit: but actually, wouldn't there still be gravitational time dilation for the clock on Earth in a "near inertial frame", so they wouldn't tick at the same rate?) So, this doesn't really seem to have anything to do with the equivalence in local observations between being at rest on Earth and accelerating in space at 1G.

Yes, you're right. Trying to get an equivalence scenario gets complicated. Given this near inertial Earth comoving frame, a clock (C1) a million miles from an Earth ground clock (C2) will tick faster by the gravitational redshift amount of C2. However, trying to compare C1 to a 1 g accelerating clock (C3) runs up against the inequivalence between this psuedo gravity and the Earth's field (uniform, non-decreasing with distance). There will be some unique distance between C3 an C1 such that when C3 is stationary with respect to C1 but accelerating at 1 g, C1 will tick faster than C3 by the same amount as it ticks faster than C2. However, this distance will be much less than a million miles. The upshot is simply what you said: it doesn't make sense to analyze this scenario using the equivalence principle.

 

[PAllen]

Does that mean when accelerating, direction affects time dilation?

The issue is that you are accelerating in spacetime curved by earth. There is a preferred direction: towards earth. In this scenario, tangential and radial acceleration do not have identical effects.

 

[rede96]

The issue is that you are accelerating in spacetime curved by earth. There is a preferred direction: towards earth. In this scenario, tangential and radial acceleration do not have identical effects.

Ok.

So if I am in a ship somewhere in space where there are no gravitational forces, the laws of physics are the same for both me accelerating at 9.81g and you being in gravitational field of 9.81g on earth. Is that right?

So if I drop a ball from 10 feet, it will take the same time to hit the floor for me as it would for you. (That is what I understood from the equivalence principle.)

I can deduce from that the two clocks would run at the same rate?

 

[JesseM]

So if I drop a ball from 10 feet, it will take the same time to hit the floor for me as it would for you. (That is what I understood from the equivalence principle.)

This is true.

I can deduce from that the two clocks would run at the same rate?

Same rate relative to what coordinate system? You can't compare clocks at different locations without a coordinate system covering both regions that has its own simultaneity convention. Think of it this way, we also know that the local laws of physics seen within different ships moving inertially would be the same as one another, so a ball tossed at the wall with the same locally-measured force would take the same amount of time to hit it as measured by local clocks within two ships moving inertially--but obviously it's not true that clocks in ships moving inertially at different speeds relative to some frame would tick at the same rate relative to coordinate time in that frame!

 

[PAllen]

Ok.

So if I am in a ship somewhere in space where there are no gravitational forces, the laws of physics are the same for both me accelerating at 9.81g and you being in gravitational field of 9.81g on earth. Is that right?

So if I drop a ball from 10 feet, it will take the same time to hit the floor for me as it would for you. (That is what I understood from the equivalence principle.)

I can deduce from that the two clocks would run at the same rate?

Really, JesseM is on target here. The local measurements in the two cases are identical. However, comparing clocks over large distances is not easily amenable to equivalence argements.

A clock far from Earth (nothing else around) would run faster than a clock on earth. There is some *speed* at which the far way clock would run the same rate as the earthbound clock.

The analogy of acceleration to gravity in this case is quite tricky. Imagine you do some amount of work to move a clock from ground to 'far away'. Now imagine a very long rocket at 1 g (or some other acceleration). Imagine lifting a clock from the back end of the rocket to a 'height' along the rocket where you have done the same work. Then the higher rocket clock will be faster compared to clock at the bottom of the rocket by the same amount as the distant clock is compared to the earthbound clock.

 

[rede96]

This is true.

Same rate relative to what coordinate system? You can't compare clocks at different locations without a coordinate system covering both regions that has its own simultaneity convention.

Sorry JesseM, I really don't understand what that means.

Think of it this way, we also know that the local laws of physics seen within different ships moving inertially would be the same as one another, so a ball tossed at the wall with the same locally-measured force would take the same amount of time to hit it as measured by local clocks within two ships moving inertially--but obviously it's not true that clocks in ships moving inertially at different speeds relative to some frame would tick at the same rate relative to coordinate time in that frame!

But if the two ships were at rest wrt each other, the clocks would tick at the same rate. Also, if two ships were accelerating at the same rate, their clocks would tick at the same rate.

As gravity and acceleration are equivalent, if I am acceleration at 1G and you are in a gravitational field of 1G, then the clocks must tick at the same rate.

So to make that relative, if I was at rest wrt to you on earth, my clock would tick faster than yours as I am not in a gravitational field.

So If I accelerate to 1G, wouldn't this equalise the difference? For the time I was accelerating, I would have aged at the same rate as you.

I can't think of one right now, but there would probably a thought experiment to demonstrate this no?

 

[rede96]

Really, JesseM is on target here. The local measurements in the two cases are identical. However, comparing clocks over large distances is not easily amenable to equivalence argements.

A clock far from Earth (nothing else around) would run faster than a clock on earth. There is some *speed* at which the far way clock would run the same rate as the earthbound clock.

The analogy of acceleration to gravity in this case is quite tricky. Imagine you do some amount of work to move a clock from ground to 'far away'. Now imagine a very long rocket at 1 g (or some other acceleration). Imagine lifting a clock from the back end of the rocket to a 'height' along the rocket where you have done the same work. Then the higher rocket clock will be faster compared to clock at the bottom of the rocket by the same amount as the distant clock is compared to the earthbound clock.

I think I get what you are saying.

I've got to go right now, but I'll have think and see if it sinks in!

Thanks all for the help!

 

[JesseM]

Sorry JesseM, I really don't understand what that means.

I'm saying that all statements about the "rate" a clock is ticking are based on comparing tick rate to coordinate time in some coordinate system--for example, if one tick has a coordinate t=0 seconds and the next has coordinate t=2 seconds, then the clock is ticking once every two seconds in that coordinate system. And are you aware of the relativity of simultaneity, which says different inertial frames (and also non-inertial frames) disagree about what events are simultaneous? So for example if two clocks A and B show the same time of "0 seconds" when they pass each other at 0.6c, then in A's frame the event of A showing a time of "20 seconds" happens at t=20 and the event of B showing a time of "16 seconds" also happens at t=20, so these events are simultaneous in this frame, and B is therefore ticking at only 20/16 = 0.8 the rate of A in this frame. But in B's rest frame, the event of A showing a time of "20 seconds" instead happens at t'=25, and the event of B showing at time of "25 seconds" also happens at t=25, so these events are simultaneous in this frame, and A is therefore ticking at only 25/20 = 0.8 the rate of B in this frame. So you see that the two frames disagree about which clock is ticking at a faster rate, and this disagreement is a consequence of the fact that they disagree about which time on B's clock was simultaneous with a given reading like "20 seconds" on A's clock.

But if the two ships were at rest wrt each other, the clocks would tick at the same rate.

Only relative to an inertial frame, you could have a non-inertial frame where they weren't ticking at the same rate, if this frame defined simultaneity differently. And in general relativity there is no "preferred" set of frames to use for large regions of curved spacetime like with inertial frames in special relativity--you can construct coordinate systems in any arbitrary way you like and the laws of general relativity will still work in these coordinate systems, see the discussion of "diffeomorphism invariance" on this page.

Also, if two ships were accelerating at the same rate, their clocks would tick at the same rate.

Only if they also started out with the same initial velocity in the frame where they had the same acceleration.

As gravity and acceleration are equivalent, if I am acceleration at 1G and you are in a gravitational field of 1G, then the clocks must tick at the same rate.

They tick at the same rate relative to local inertial frames in the immediate neighborhood of each clock where that clock starts out at rest, but that's no reason to conclude they should tick at the same rate in a larger coordinate system which covers a region of spacetime large enough to contain both clocks. Similarly if my ship and your ship are moving inertially relative to one another in deep space, the rate that my clock ticks relative to my inertial rest frame is the same as the rate your clock ticks relative to your inertial rest frame, but that's no reason to think both our clocks tick at the same rate in a single inertial frame which contains both of us.

 

[ZealScience]

I am just trying to understanding of the equivalence principle and was hoping someone would be kind enough to sense check my thoughts below.

1) If I was in a rocket ship accelerating at a rate of 1g relative to the earth, the clock on my ship would tick at the same rate as a clock on the earth.

2) If I was to then decelerating at the rate of 1g (i.e. negative acceleration) wrt to the earth, although the force I would feel would be repelling me wrt to the earth, the two are still equivalent. So a clock on my ship would tick at the same rate as a clock on earth.

3) If I was to travel in a circle around the Earth such that the net force I felt was equal to 1g, although the force I would feel would be repelling me wrt to the earth, the two are still equivalent. So a clock on my ship would tick at the same rate as a clock on earth.

4) If the above are true, why is there no time dilation or length contraction, as I am moving with wrt the earth?

Thanks!

I am confused, if you are moving respect to earth, you are moving through the G field or curvature of space due to a change in area enclosed by a loop. Then there would be a change in time, so why not?

Also, if you are circum navigating arround the earth, you would detect different curvature comparing to that on surface of earth, so the space time also varies.

Lastly, in my pers pective ,if you are accelerating (whether in a field or not), just use Lorentz transformation in special relativity, you will find a time change.

 

[rede96]

Had a chance to reflect and can see where I was getting confused. I guess my original post didn't have too much to do with the equivalence principle and was more to do with understanding time dilation.

So, to put my question in a different way...

In SR, if two frames are at rest wrt each other, then time passes at the same rate. e.g. if there were two space stations at rest wrt to each other and I sent my twins to live on them, one on each, for a period of 10 years and then return, they would have both aged the same. (Ignoring the traveling to and from me.)

In GR, if two solar systems are at rest wrt to each other, each with a planet of equal gravity, (assuming similar orbits too.) and I send my twins to live on the planets for 10 years, (again one on each)when they return they would have both aged the same.

If I send one twin to a solar system with a planet of gravity of 1G that orbits a sun. I send the other to a spaceship which circles around a centre point which is at rest wrt to the sun in the other solar system and where the equivalent force of this is 1G, when they returned would they have aged the same? (Again, ignoring the initial traveling to and from me.)

From what I understood of the previous replies, the answer is probably not. As the twin that was in the spaceship would have a faster 'orbit' than the twin on the planet in the other solar system, in order for him achieve a force of 1G. So he would age less.

So, the question is…

Is there a way in which one twin could be accelerated to a feel a force of 1G and the other to be in a gravitation field of 1G and them both age the same when they returned?