Prove that acceleration and gravity are really the same?

[stevendaryl]

It’s not that I am totally against the idea, it’s just that I would like a reasonable explanation.

An explanation for why the equivalence principle is true? Well, from the point of view of Newtonian gravity, you can explain it this way:

The only time that you "feel" acceleration is when different parts of your body are accelerated in different ways. If you are on a rocket standing on the floor, the floor is pushing up on your feet, but the rest of your body has no (external) forces acting on it. So the stress that you feel is your feet being accelerated relative to the rest of your body. Your body will compress slightly until the internal forces in your body are in equilibrium.

In contrast, gravity accelerates all parts of your body equally. So your feet are not accelerating relative to the rest of your body, and your body isn't compressed. So you feel no acceleration.
​

Einstein's explanation for the equivalence principle is different. In his theory of GR, he explains that freefall feels like inertial motion because it IS inertial motion, only inertial motion in curved spacetime rather than flat spacetime.

 

[MikeGomez]

Here's a simple model for an accelerometer: You have a cubic box. In the middle of the box is a massive metal ball. It is held in place in the center of the box by 6 identical springs connecting the ball to each of the 6 sides of the box.

When you accelerate, the ball will move from the center. By measuring the lengths of the 6 springs, you can compute what the acceleration vector is.

Thank you. I know how an accelerometer works.

An explanation for why the equivalence principle is true?

Seriously? No. I wasn't asking for an explanation of the equivalence principle. I was asking for a better explanation (than the falling accelerometer) of why case #3 should not be called acceleration, but must be called "coordinate" acceleration.

So you honestly don't see falacy in the accelerometer example?

gravity accelerates all parts of your body equally

That's exactly why accelerometers doesn't measure the gravitational acceleration in case #3.

It's not complete rubbish

I'm still not convinced.

Anyways, thanks for your time. I don't see this going anywhere productive, so I'll just concede that case #3 is definitely not acceleration, and that great care should be taken to make the distinction that it is "coordinate" acceleration.

 

[stevendaryl]

Thank you. I know how an accelerometer works.

Then I don't understand your objection to the claim that an accelerometer can't distinguish between being stationary on a planet vsl accelerating on a rocket, or between being in freefall in a gravitational field vs. traveling inertially. It definitely CAN distinguish between accelerating due to gravity and accelerating due to any other force.

Seriously? No. I wasn't asking for an explanation of the equivalence principle. I was asking for a better explanation (than the falling accelerometer) of why case #3 should not be called acceleration, but must be called "coordinate" acceleration.

Because an accelerometer measures no acceleration. That's a coordinate-independent definition of acceleration. In contrast, coordinate acceleration depends on having a coordinate system to measure acceleration relative to. That's why it's called "coordinate acceleration".

So you honestly don't see fallacy in the accelerometer example?

No, I have no idea what you are talking about when you say there is a fallacy involved.

I'm still not convinced.

Convinced of what?

 

[LitleBang]

Wouldn't it be nice if the standard model people would show why gravity(graviton) appear to be identical with inertia?

 

[Dale]

I was asking for a better explanation (than the falling accelerometer) of why case #3 should not be called acceleration, but must be called "coordinate" acceleration.

So you honestly don't see falacy in the accelerometer example?

I also fail to see the fallacy. It is simply a matter of definition. By definition "coordinate acceleration" is the second derivative of position wrt time. By definition "proper acceleration" is the second covariant derivative along the worldline (physically the acceleration as measured by a correctly functioning accelerometer). There are cases where the two definitions coincide and cases where they do not coincide, nothing fallacious about that.

You can certainly use the unqualified term "acceleration" how you choose, but if the case under discussion is one where they do not coincide then you invite confusion and ambiguity.

 

[MikeGomez]

No, I have no idea what you are talking about when you say there is a fallacy involved.

Here’s a picture.

Accelerometer A is at rest in the lab frame produces no reading (no compression or tension in the springs.

Accelerometer B is being accelerated from the outside of the box (as designed) and therefore gives a reading.

Accelerometer C is being accelerated with the same force as Accelerometer B, but in a manner in which the apparatus was not designed for. I have opened the lid of the box, and I have accelerated it evenly so that there is no tension/compression in the springs. I couldn’t draw it very well, but you can imagine, at least in principle, that every unit of mass is accelerated.

I believe that at a fundamental level, every unit of mass in accelerometer 'C' gains momentum in precisely the same way as it would by gravity in free fall. I could be mistaken, but that is how I currently understand it. So when I say "Accelerating due to gravity" (coordinate-acceleration in free fall) is equivalent to an "accelerating chest", that is my reasoning.

On the other hand, what would you think if I were to tell you that accelerometer C is not accelerated because it shows no reading? Doesn’t that sound silly? So why should that same situation (the misuse of the apparatus) suddenly start making sense simply because the accelerometer is in freefall?

Hopefully that explains why I believe the accelerometer in freefall explation is bunk. If not, oh well.

 

[MikeGomez]

...
By definition "coordinate acceleration" is the second derivative of position wrt time. By definition "proper acceleration" is the second covariant derivative along the worldline
(physically the acceleration as measured by a correctly functioning accelerometer). There are cases where the two definitions coincide and cases where they do not coincide, nothing fallacious about that.

O.k. Thank you for that. I guess in my example, case 'C' is the case where they coincide?

 

[stevendaryl]

Here’s a picture.

Accelerometer A is at rest in the lab frame produces no reading (no compression or tension in the springs.

That would be true if your lab were floating in space. Here on Earth, an accelerometer would measure compression. It would indicate upward acceleration. Of course, since g is roughly constant on the surface of the Earth, a practical accelerometer would be calibrate to subtract off this effect, but it is certainly measurable.

Accelerometer B is being accelerated from the outside of the box (as designed) and therefore gives a reading.

Accelerometer C is being accelerated with the same force as Accelerometer B, but in a manner in which the apparatus was not designed for. I have opened the lid of the box, and I have accelerated it evenly so that there is no tension/compression in the springs. I couldn’t draw it very well, but you can imagine, at least in principle, that every unit of mass is accelerated.

I believe that at a fundamental level, every unit of mass in accelerometer 'C' gains momentum in precisely the same way as it would by gravity in free fall. I could be mistaken, but that is how I currently understand it. So when I say "Accelerating due to gravity" (coordinate-acceleration in free fall) is equivalent to an "accelerating chest", that is my reasoning.

Okay, I would agree that accelerating due to gravity is equivalent to an accelerating chest in which it has been carefully arranged so that every single particle in the box accelerates in exactly the same manner. But that's not the normal way of accelerating things. If you accelerate something by pushing on one end, that end will accelerate more than the other end. If you accelerate something by pulling on one end, that end will accelerate more. If you accelerate a charged particle by an electric field, then oppositely charged particles will be accelerated in the opposite direction.

There is no way to accelerate all parts of an object equivalently OTHER than using gravity. You might be able to do it approximately, though.

On the other hand, what would you think if I were to tell you that accelerometer C is not accelerated because it shows no reading? Doesn’t that sound silly?

Frankly, exactly the opposite. It seems silly to insist that something is accelerating, even though there is no way to measure it. Your case C is a very special case. It's not something that can be arranged except through very special forces that treat all phenomena in the same way. (And it really has to be ALL phenomena; even the path of a light beam is accelerated by gravity. You can't arrange that by any other means, as far as I know).

So why should that same situation (the misuse of the apparatus) suddenly start making sense simply because the accelerometer is in freefall?

I wouldn't use the phrase "misuse of the apparatus". It's not a matter of misusing the apparatus, it's a matter of how to interpret its reading. You would like to interpret it as measuring coordinate acceleration. It doesn't measure that, except in special circumstances. It measures proper acceleration, which is acceleration relative to freefall.

That's true in general. Devices don't measure coordinate-dependent quantities. You have to do an act of interpretation to interpret their measurements as saying something about coordinates.

Hopefully that explains why I believe the accelerometer in freefall explation is bunk. If not, oh well.

The equivalence of freefall to inertial (acceleration-free) motion is just true (to the extent that variability of gravity with position can be ignored). And that equivalence was instrumental for Einstein to develop General Relativity.

So to me, it makes no sense to call something bunk when it is both truth and important.

But, anyway, regardless of whether you like it, that is what is meant by "the principle of equivalence". So your way of describing the principle of equivalence is not what "the principle of equivalence" means.

 

[stevendaryl]

O.k. Thank you for that. I guess in my example, case 'C' is the case where they coincide?

No. The two types of acceleration (coordinate vs. proper) coincide when there is no gravity. Your case C is a case where they DON'T coincide.

 

[DrGreg]

MikeGomez,

Acceleration has to be measured relative to something. So, to avoid all ambiguity, whenever anyone talks of "acceleration", they need to specify relative to what. Some of the confusion in this thread has been that the "what" hasn't always been explicitly mentioned and different people have been assuming different "what"s.

After you throw a ball in the air, you can say it has a (more-or-less) constant acceleration g downwards relative to the ground and the ground has an acceleration of zero relative to the ground. Or you can say that ground has a (more-or-less) constant acceleration g upwards relative to the ball and the ball has an acceleration of zero relative to the ball. They are both equally valid descriptions.

There is a special case when the reference object (that we measure relative to) is moving inertially, and is momentarily moving at the same speed as the object being measured. In that case we call this "proper acceleration". In Newtonian physics, the ground was taken to be inertial and so the falling ball would have non-zero "Newtonian proper acceleration". Einstein's revolutionary idea was to take the ball to be inertial and the ground to have a non-zero (relativistic) proper acceleration. That last sentence is really another way to describe GR's equivalence principle.

It turns out, using the GR definition of inertial, that accelerometers always measure (relativistic) proper acceleration whatever object you attach them to, whereas in Newtonian physics accelerometers fail to measure "Newtonian proper acceleration" when used in a gravitational field (unless you "calibrate out" the effects of gravity).


I put the phrase "Newtonian proper acceleration" in quotation marks because I don't think anyone actually uses the term "proper acceleration" in Newtonian physics.

 

[Dale]

O.k. Thank you for that. I guess in my example, case 'C' is the case where they coincide?

I am not 100% clear on your example C, but in simplified terms if a force is proportional to mass then it can be geometrized. That means that it can be lumped into the same category as the centrifugal and Coriolis forces in a rotating reference frame, and canceled out by a coordinate transform in the same way. The proper acceleration is given by a covariant derivative, so it cannot be canceled by a coordinate transform.

So, in your scenario C, if you intend the force acting on the accelerometer to be proportional to mass then you can find a coordinate system where it is locally canceled out. In this coordinate system proper acceleration and coordinate acceleration coincide locally. In other coordinate systems they would not coincide.

 

[A.T.]

The apparatus is used in this example to take a measurement for which it was not designed.

The point is that you can't design a local measurement of coordinate acceleration due to gravity. Maybe this video will help you:

https://www.youtube.com/watch?v=QSIuTxnBuJk

 

[davenn]

Thanks AT

interesting video ... went on to watch a few of the others as well :)

Dave

 

[HALON]

I also found the video good value. It made me think of this question:

Body A at rest on planet X’s surface experiences gravity equivalently to body B undergoing acceleration not near any major sources of gravity in open space.

What is the formula used to convert B’s acceleration to X’s mass?

 

[Nugatory]

I also found the video good value. It made me think of this question:

Body A at rest on planet X’s surface experiences gravity equivalently to body B undergoing acceleration not near any major sources of gravity in open space.

What is the formula used to convert B’s acceleration to X’s mass?

There's no single answer, as the necessary mass also depends on the radius of the planet. You want the gravitational force from X on A to be sufficient to produce the acceleration that B is experiencing. Newton's law for gravitational force $F=Gm_1m_2/r^2$ and $F=ma$, plus a little algebra, will get you to the formula.

 

[HALON]

Here's how I entertained the old standard. I knew it of course.

In $F=m_1a$ and $F=Gm_1m_2/r^2$

let $m_1=1$, let $a=1$, and let $r=1$. The mass of planet X is $m_2$ but let's call it $m_X$

Then to find $m_X$ we get

$m_X = 1/G$

This particular answer is nice and neat. It's just a mnemonic.

 

[Nugatory]

This particular answer is nice and neat. It's just a mnemonic.

But rather unhelpful unless you know the value of $G$ in units in which $r$ and $a$ are both equal to one

 

[Wes Tausend]

...

Interesting thread!
Equivalence is Thee Key to GR.

For the gentlemen that found difficulty in understanding the Equivalence principle, it may be helpful to imagine the situation of three clowns drifting in power-off-mode inside a free-floating spaceship in outer space, far from any major gravities.

Imagine two of them, Moe and Larry, are moving hand-to-hand along a longitudinal "fire" pole that runs dead-center the length of the spaceship just for such weightless ambulatory purposes. They have already traveled 100 feet along its length from the back towards the front of the ship. The ship is so massive that a couple of humans pulling on it to propel themselves has negligable inertial effect on the ship.

About this time, the third clown, Curly (always the darn fool) unfortunately started the powerful, poorly tuned spaceship engines in notch one throttle, and the back-firing spaceship suddenly accelerated with a bang up to 32 ft/sec/sec and held there. Moe and Larry lost their handhold and were left floating as the rear bulkhead of the ship approached rapidly. Moe happened to be carrying an accelerometer that had been faithfully reading zero, at least until the rear bulkhead crashed into them. The accelerometer needle bounced wildly high from the impact for a moment, and then settled down to the ship acceleration of 32 ft/sec/sec. The accelerometer lived... and Moe and Larry were livid too.

Curly had just caused them an accident equivalent to falling 100 feet on earth. When the engines fired, Moe and Larry didn't initially accelerate until they hit the floor (the bulkhead hit them), which is really a more correct Einsteinian-like Equivalence definition of an earthern episode of freefall. For the record, in spite of two broken legs, when he got the chance, Moe slapped Curly up and poked him in the eyes for the latest dumb stunt of course.
====================================================================

Gravity is a force which causes acceleration. Gravity is not the same as acceleration.

If gravity were acceleration, then the radius of Earth would have to be expanding with an acceleration of 9.8 m/s^2, and we would have been very confused when we measured it from higher altitudes.

Actually, in both cases, Earth atomically expanding or gravitational attraction, higher altitudes would register less acceleration, just as one weighs less at the top of a mountain. It is a matching qualitive property only, and a little difficult to understand at first, but here is the reasoning.

Suppose Captain Kirk and Scotty always happened to weigh exactly the same as long as they weighed in side-by-side on the same bath scale on Earth. Then one day Kirk took the scale with and climbed a nearby mountain that was 1/2 mile high and weighed himself again. He called Scotty and told him he weighed slightly less at the top. Scotty, finding this interesting, asked Kirk to take a picture. Kirk took a selfie including the scale dial for proof and sure enough, later the picture showed he weighed less at the top. But back at the bottom, side-by-side he again weighed the same as Scotty.

The next day both men weigh in and they again weigh the same of course. But today is the start of their trip to a new quadrant of space, and they climb aboard the latest Enterprise spaceship. It is a big cigar-shaped craft, 1/2 mile long. Kirk goes to the head of the ship on the bridge. Scotty remains in the rear of the ship in the engine room, and they take off. For comfort with simulated gravity, they soon drop to 32 ft/sec/sec acceleration after initial take-off. They both end up standing on separate bulkheads behind them, the one nearest to themselves, Kirk in front and Scotty in the rear. They are 1/2 mile apart.

Under constant acceleration, Kirk will find that he weighs slightly less than Scotty again. The reasoning is that as the ship attains higher and higher speeds, it also begans to continuously foreshorten according to Special Relativity. So as the ship gets shorter, either Kirk is relatively accelerating slower than Scotty, or Scotty is relatively accelerating faster than Kirk. Either way Kirk experiences less acceleration than Scotty, proven if he checks himself against the trusty ol' spring-loaded bathroom scale.

Not only that, but Kirks precision watch will run faster than Scotty's identical timepiece, on the same ship!

Now just because both the mountain and ship are exactly the same half-mile length does not mean Kirks weight loss will necessarily be the same percentage in both cases. It is my understanding (according to a better mathematician on these forums than I) that the two effects, though similar, are not quantitively the same. They are, however qualitively very similar. And as well they should be... after all Equivalence is equivalent to a very great degree, good enough for Einstein to base his new theory of GR upon.

Wes
...

 

[rcgldr]

Back to the original post, the equivalence principle is based on local observation, where the observer is either in a gravitational field or being accelerated. In the case of an "external" observer, such as an observer in an inertial frame where distant stars can be used as a frame of reference, then that observer can determine if an observed object is being affected by gravity and/or by acceleration.

 

[Wes Tausend]

Back to the original post, the equivalence principle is based on local observation, where the observer is either in a gravitational field or being accelerated. In the case of an "external" observer, such as an observer in an inertial frame where distant stars can be used as a frame of reference, then that observer can determine if an observed object is being affected by gravity and/or by acceleration.

Unless the acceleration effect was pervasive throughout the universe.

Is there a way by which we can prove that acceleration and gravity are really the same?

Yes, back to post #1 instead of post #2. When #2 immediately, and mistakenly, implied doubt, I thought it was quite important to point out to the OP that the Equivalence principle is much more equivalent than most folks think.

For the OP, it is not possible to "prove that acceleration and gravity are really the same", only point out that, under some conditions, if all is considered, it becomes impossible to tell the difference. They are at least identical twins. It has been merely our convenient choice to sometimes regard them as separate individuals. Einstein was not the first to realize the remarkable equivalence (Newton and others certainly did too), but he was the first to postulate that they are equivalent so that he could treat the "similarity" mathematically in the new light of SR.

To postulate Equivalence, Einstein imagined a couple of scientists being drawn up in an enclosed chest, and not being able to distinguish whether they were in a gravitational field or not. Although Einstein never stated so, one of the tests could have been that two objects, when dropped simultaneously, might fall parallel, or might fall in such a way as to move closer together on the way down. In a real gravitational system they would fall closer as they dropped (towards earthen center). An expanding globe would allow for this to be indistinguishable as long as the scientists, and chest, and space between them equally expanded also. The psuedo-gravities would be even more identical in this case over the plain chest example. The hapless scientists would not realize the wholesale change in size, only the resulting inertial effect they would surely assume is that of gravity.

This expansion principle brought up by post #2 arose before. Perhaps with great foresight, the great French mathematician, and philosopher of science, Poincaré, once imagined that the world could change dimension overnight, and no one could measure the difference the next day. Or perhaps the next moment. Measure no, but what about the inertia?

Wes
...

 

[Nugatory]

Unless the acceleration effect was pervasive throughout the universe.

Which isn't possible (and I presume that's why you put the smiley in there).

An infinite charged sheet at constant potential would fill the entire universe with a constant electrical field, but there's no analogous gravitational solution.