Is Gravity inertia, acceleration or curvature in GR?

[fog37]

Hello Forum,

I have read Einstein famous thought experiments about the elevator.

1) Being inside an elevator accelerating upward in absence of a gravitational field is equivalent to being inside the same elevator at rest inside a homogeneous gravitational field.
2) An elevator in free fall, i.e. accelerating downward at 9.8 m/s^2 inside a gravitational field is the same as being in the absence of the gravitational field

I read that gravity is the same as inertia. In some other books gravity is said to be equivalent to acceleration since acceleration (an accelerated frame of reference) can simulate gravity and free fall can also simulate the absence of gravity.

How do we tie these ideas that gravity is acceleration (or inertia?) to the fact that gravity is interpreted and geometrized as the curvature of space time (curvature is caused by mass) in special relativity? Objects follow geodesics in the curved spacetime...

thanks,
fog37

 

[mfb]

I read that gravity is the same as inertia.

Where did you read that? It is not.

In some other books gravity is said to be equivalent to acceleration since acceleration (an accelerated frame of reference) can simulate gravity and free fall can also simulate the absence of gravity.

With that logic, a centrifuge would be equivalent to gravity as well? I see a large difference between a man-made machine and a natural effect like gravity.

Objects follow geodesics in the curved spacetime

That is the point. If you stand on the surface of Earth, you do not follow a geodesics, and someone who does seems to accelerate relative to you.

 

[fog37]

Many books, when discussing the elevator experiments, equate gravity to acceleration... that is why I am confused...

 

[fog37]

Related question:

According to Newtonian mechanics motion is relative only in inertial frames of reference and not in accelerated frames of reference. But if someone is inside a noninertial frame, couldn't he think that it is the other frame that is accelerating toward his frame?

Why do we realize that we are moving when we are in an accelerated frame of reference? By introducing apparent, fictitious forces, we could simply think that are at rest and there are forces in action counterbalancing each other. For instance, inside an accelerating car, we feel pushed against the car seat...

Einstein later proves that motion is relative in ALL reference frames...why?

 

[A.T.]

By introducing apparent, fictitious forces, we could simply think that are at rest and there are forces in action counterbalancing each other

Since we had to introduce fictitious forces to makes Newtons 2nd Law work, we know that the frame is not inertial.

Einstein later proves that motion is relative in ALL reference frames...why?

Relativity distinguishes inertial and non-inertial frames as well.

 

[mfb]

Many books, when discussing the elevator experiments, equate gravity to acceleration... that is why I am confused...

They are not equal. They can have similar effects.
Running against a wall can have a similar effect as running against a truck. That does not mean buildings and trucks are the same thing. They just share a few properties (e.g. you shouldn't run into them).

According to Newtonian mechanics motion is relative only in inertial frames of reference and not in accelerated frames of reference.

Motion is always relative to a reference frame, independent of its acceleration.

Why do we realize that we are moving when we are in an accelerated frame of reference?

We do not. We just note that we are accelerating. The car seat pushes you, therefore you accelerate.

 

[alw34]

I read that gravity is the same as inertia. In some other books gravity is said to be equivalent to acceleration since acceleration (an accelerated frame of reference) can simulate gravity and free fall can also simulate the absence of gravity.

Roughly speaking, with caveats, ok. The original equivalence principle, as described by Einstein, concluded that free-fall and inertial motion were physically equivalent. But our understanding has improved over time and so there are several versions of the 'equivalence principle'. Several are described here: https://en.wikipedia.org/wiki/Equivalence_principle

...gravity is interpreted and geometrized as the curvature of space time (curvature is caused by mass) in special relativity? Objects follow geodesics in the curved spacetime...

In general, gravity in the real world does involve [special] 'curvatures' of space time, but the elevator is an idealized LOCALIZED thought experiment where, as you implied in your post, gravity is uniform, meaning spacetime is not curved in that special sense. It's assumed to be 'locally flat' or uniform gravitationally.

In the real world, gravitational curvature is characterized by tidal effects...stretching and compression...as a result of the special curvature of gravity. That special curvature is captured in Einsteins General Relativity via his stree energy moementum tensor. That special curvature is not in the idealized elevator setting.

Einstein deduced that free fall in a gravitational field is actually inertial motion, motion along geodesics. Objects in free-fall do not experience acceleration, as in the Newtonian perspective, but rather weightlessness. In Newtonian physics, bodies move at constant velocity in straight lines. In relativity, the world line [path] of an inertial particle or pulse of light is as straight as possible in space and time, along a geodesic as you noted.

With proper acceleration, you feel a force...that is, an accelerometer would register it. But in gravitational free fall, you feel no force, so Einstein would say you are not accelerating.

 

[alw34]

Why do we realize that we are moving when we are in an accelerated frame of reference?

Where you 'accelerate but do not move' is sitting in your chair reading this. You feel a force pushing against your backside, hence you are accelerating.

 

[alw34]

Being inside an elevator accelerating upward in absence of a gravitational field is equivalent to being inside the same elevator at rest inside a homogeneous gravitational field.

So with the previous posts, you should now be able to see the above is not necessarily 'the same' but some effects may be similar. Also, the Unruh effect is an accepted phenomena in the accelerating elevator but not the 'at rest' case.

 

[mfb]

Why do we realize that we are moving when we are in an accelerated frame of reference?

As I said before, we do not. We can note the acceleration (more specifically, the force accelerating us).

Also, the Unruh effect is an accepted phenomena in the accelerating elevator but not the 'at rest' case.

If the accelerator is at Earth, you have to be in free fall to not see Unruh radiation. Standing in an accelerator you see it (well, at least theoretically).

Edit: Fixed quote reference

 

[alw34]

If the accelerator is at Earth, you have to be in free fall to not see Unruh radiation. Standing in an accelerator you see it (well, at least theoretically).

oops...I think you mean sitting in my seat I should see Unruh radiation...of course, you are right...thanks

but, that first 'quote' is not mine, but the op...for unknown reasons I could not obtain a 'reply' caption...

 

[MikeGomez]

I read that gravity is the same as inertia.
fog37

Where did you read that? It is not.

Einstein said it many times.

With that logic, a centrifuge would be equivalent to gravity as well?

Exactly.

 

[mfb]

Einstein said it many times.

Do you have a source?
He said many things that use those words in a different order and with a different meaning, but I'm quite sure that he did not say.

 

[MikeGomez]

Do you have a source?
He said many things that use those words in a different order and with a different meaning, but I'm quite sure that he did not say.

Blue highlighting are by me.

From Einstein’s 1918 paper: On the Foundations of the General Theory of Relativity
http://einsteinpapers.press.princeton.edu/vol7-trans/49

“Inertia and gravity are phenomena identical in nature.”
- Albert Einstein

In a letter Einstein wrote in reply to Reichenbacher
http://einsteinpapers.press.princeton.edu/vol7-trans/220

“I now turn to the objections against the relativistic theory of the gravitational field. Here, Herr Reichenbacher first of all forgets the decisive argument, namely, that the numerical equality of inertial and gravitational mass must be traced to an equality of essence. It is well known that the principle of equivalence accomplishes just that. He (like Herr Kottler) raises the objection against the principle of equivalence that gravitational fields for finite space-time domains in general cannot be transformed away. He fails to see that this is of no importance whatsoever. What is important is only that one is justified at any instant and at will (depending upon the choice of a system of reference) to explain the mechanical behavior of a material point either by gravitation or by inertia. More is not needed; to achieve the essential equivalence of inertia and gravitation it is not necessary that the mechanical behavior of two or more masses must be explainable as a mere effect of inertia by the same choice of coordinates. After all, nobody denies, for example, that the theory of special relativity does justice to the nature of uniform motion, even though it cannot transform all acceleration-free bodies together to a state of rest by one and the same choice of coordinates.”
- Albert EinsteinFrom Albert Einstein’s book: The Meaning of Relativity, pg 58

“… In fact, through this conception we arrive at the unity of the nature of inertia and gravitation. For according to our way of looking at it, the same masses may appear to be either under the action of inertia alone (with respect to K) or under the combined action of inertia and gravitation (with respect to K’). The possibility of explaining the numerical equality of inertia and gravitation by the unity of their nature gives to the general theory of relativity, according to my conviction, such a superiority over the conceptions of classical mechanics, that all the difficulties encountered must be considered as small in comparison with the progress.”
- Albert Einstein

 

[mfb]

that the numerical equality of inertial and gravitational mass must be traced to an equality of essence

That is a completely different statement. And I guess the other quotes are meant in the same way.

 

[alw34]

“Inertia and gravity are phenomena identical in nature.”

Hard to tell exactly what that means, but it sure seems early 1900's classical inertia and Einstein's new gravitational mass are different. For one, the latter 'phenomena' does not operate instantaneously. Is acceleration as geodesic deviation in GR 'identical' to classical acceleration?

As mfb says, 'equality of essence' is different, depending on your definitions, but you'd never know that from the following:

In the Wikipedia article I already linked to,
https://en.wikipedia.org/wiki/Equivalence_principle#Modern_usage

there are a number of 'equivalence principles' discussed and if the article is historically accurate there seem to be some uncertainties about exactly how they are related. In addition, I'd suspect Einstein might have used different descriptions depending on whether he was talking to fellow scientists or the general public.

I thought I'd see if I could find some properly described differences between inertia and gravity in a source using current language and norms, but here
https://en.wikipedia.org/wiki/Inertia#Interpretations

once again language and understanding seems to be sloppy and also meanings vary over time.

Here's an example from that article, and I'd argue some of the following is at best rather self contradictory:

"...No physical difference has been found between gravitational and inertial mass in a given inertial frame. In experimental measurements, the two always agree within the margin of error for the experiment. Einstein used the fact that gravitational and inertial mass were equal to begin his general theory of relativity in which he postulated that gravitational mass was the same as inertial mass.... Dennis Sciama later showed that the reaction force produced by the combined gravity of all matter in the universe upon an accelerating object is mathematically equal to the object's inertia [1], but this would only be a workable physical explanation if by some mechanism the gravitational effects operated instantaneously.

At high speeds, relativistic mass always exceeds gravitational mass. If the mass is made to travel close to the speed of light, its "inertial mass" (relativistic) as observed from a stationary frame would be very great while its gravitational mass would remain at its rest value, but the gravitational effect of the extra energy would exactly balance the measured increase in inertial mass...."

To explain an observation you have to explain from what inertial frame the observation is made. Hard to tell what those boldface statements are intended to mean, but I don't think the velocity of a mass relative to an external inertial frame changes the Einstein stress energy momentum tensor of that mass, the gravitational source.That's determined from the frame of the mass. How could you even "observe gravitational mass" from such an external inertial frame? What I think is being described is that an orbit or path of a high speed mass will be different from the gravitational path of an identical but slow moving mass.

 

[Nugatory]

Is acceleration as geodesic deviation in GR 'identical' to classical acceleration?

This question is poorly framed, as the physics is in the math and not in mushy natural-language words - that you felt the need to put scare-quotes around the word 'identical' suggests that you're aware of the problem with the question yourself.

But with that said, and for some reasonable interpretations of the mushy language, the answer would be "locally, yes".

 

[alw34]

This question is poorly framed, as the physics is in the math and not in mushy natural-language words -

exactly...that's been the issue for the last half dozen or more posts...

 

[stevendaryl]

Einstein said it many times.

No, he didn't say that gravity is the same as inertia. What his equivalence principle says is that the force of gravity is the same (or is the same sort of force) as the "inertial forces" that you feel when accelerating.

 

[Ibix]

From Albert Einstein’s book: The Meaning of Relativity, pg 58

That quote is on p60 in my edition, but in any case is just a few pages into his discussion of General Relativity. From its context I think it's clear he is talking about inertial and gravitational masses being the same, rather than inertia and gravity in general.

Please note that the book is a translation from German of a lecture series Einstein gave in 1921. There may have been a bit of change of terminology in the intervening 95 years.

 

[PeterDonis]

What his equivalence principle says is that the force of gravity is the same (or is the same sort of force) as the "inertial forces" that you feel when accelerating.

Even this isn't quite right. A better statement would be that the force you feel when you are at rest in a gravitational field is the same sort of force as the inertial force you feel when accelerating. But the force you feel when you are at rest in a gravitational field is not "the force of gravity".

 

[stevendaryl]

Even this isn't quite right. A better statement would be that the force you feel when you are at rest in a gravitational field is the same sort of force as the inertial force you feel when accelerating. But the force you feel when you are at rest in a gravitational field is not "the force of gravity".

Yes, that's better.

 

[MikeGomez]

That is a completely different statement. And I guess the other quotes are meant in the same way.

Convincing and solid counter-argument, or hand-wavy dismissal?

 

[MikeGomez]

No, he didn't say that gravity is the same as inertia.

Then how would you interpret “Inertia and gravity are phenomena identical in nature.” This is much to clear to be a German/ English translation problem.

What his equivalence principle says is that the force of gravity is the same (or is the same sort of force) as the "inertial forces" that you feel when accelerating.

Yes, he equated them when said that gravity is a fictitious force. I would say it without the qualification in parenthesis though… What his equivalence principle says is that the force of gravity is the same as the inertial forces that you feel when accelerating.

 

[PeterDonis]

What his equivalence principle says is that the force of gravity is the same as the inertial forces that you feel when accelerating.

No, that isn't what it says. See my post #21.

 

[MikeGomez]

No, that isn't what it says. See my post #21.

Understood, but I think for some people the statement “the force you feel when you are at rest in a gravitational field is not 'the force of gravity'", can cause as much confusion as clarification.

The point I was trying to make is that when Einstein wrote ground breaking theoretical papers on the equivalence of gravitational mass and inertial mass, it would have been obvious to him that this idea could easily be extended to the idea of the equivalence of gravity and inertia. If he thought this to be a mistake, he would have cautioned the reader against making this assumption, and if he believed it to be correct, he would have enforced it in his writings (which he did).

 

[PeterDonis]

I think for some people the statement “the force you feel when you are at rest in a gravitational field is not 'the force of gravity'", can cause as much confusion as clarification.

If so, the proper response is to address the confusion by explaining why the statement is true, not by muddying the waters. Even in Newtonian mechanics, the "force of gravity" is not a force you feel. Objects moving solely under gravity feel no force at all. So even in Newtonian mechanics, the force you feel standing on the Earth's surface is not "the force of gravity". It's the force of the ground pushing up on you. The "force of gravity", in Newtonian mechanics, explains why you don't fly off the Earth's surface even though the ground is pushing up on you.

when Einstein wrote ground breaking theoretical papers on the equivalence of gravitational mass and inertial mass, it would have been obvious to him that this idea could easily be extended to the idea of the equivalence of gravity and inertia.

I disagree, because the phrase "equivalence of gravity and inertia" is vague, while the phrase "equivalence of gravitational mass and inertial mass" is precise. Einstein was very careful about precision.

 

[Fervent Freyja]

This post is crazy-making, aren’t many of the arguments here based upon on a false premise, or type of bias? Einstein wrote and published in German almost exclusively. The issue looks to be in overestimating the accuracy behind the translation to primary sources/textbooks/popularized works. Isn't gathering the most accurate interpretation or understanding of what he really said dependent upon on a person’s ability to read the original-source publication in German for themselves? What Einstein really said in these above instances wasn't in English, maybe, "this translator said that Einstein said..."

“…the companion volume to the annotated, original-language documentary edition. As we have stated in all earlier volumes, these translations are not intended for use without the documentary edition, which provides the extensive editorial commentary for a full historical and scientific understanding of the source documents.”-http://einsteinpapers.press.princeton.edu/vol7-trans/7

It is important to remember that target-text translations (even if deemed a primary source) are not precise reflections of meaning from original-source publications in any case. But, particularly in this one, we have to account for the differences in publishing standards from a century ago. It had not been written with the same translating principles in mind that we have today and was aimed at a particular academic group with certain knowledge. You can’t expect such translations to be precise, even in the most technical papers, the man himself had a subtle, unique sense of humor and that makes it all the more difficult to translate or read (prominent issue encountered in his writings). There had been all sorts of media releases not long ago where he was accused of being a misogynist and what not, when much of it had been the translations/interpretations, missing real context of the culture, and important aspects of his first marriage, and his humor being misunderstood. How many of those media publishers actually went back to check his letters to find proof? I have found no counter there. I could go on...

Anyway, had he signed off on these translations, he probably would have only checked the work and not words in the paper.

Maybe the original publication should be examined in order to have it cleared up in this post -- in German? That only makes sense.

 

[pervect]

One comment that I think is worth making. "Curved" is a bit ambiguous in popular language, in the context of GR though, curvature is generally understood as meaning intrinsic curvature, such as the Gaussian curvature of a plane, see for instance https://en.wikipedia.org/wiki/Gaussian_curvature.

Einstein's elevator is *not* curved in the sense above, but is in what's called "flat" (as in not curved) "Minkowskii space-time.

Without getting too technical, we can largely equate tidal gravity with curvature. So it's common to say that Einstein's elevator has a uniform gravitational field (even though an object will weight slightly different amounts at the top and bottom of the elevator), and thus it has no tidal gravity and no curvature.

This doesn't really tell us what Gravity is - but it at least tells us what it is not. One might say that gravity, in the sense that it's used in Einstein's elevator, is mathematically represented viaa Christoffel symbols, but while this is true, I'm not sure if it helps much as a definition intended for popular audiences.

 

[PeterDonis]

extrinsic curvature, such as the Gaussian curvature of a plane

I think you mean "intrinsic", correct?

 

[stevendaryl]

Then how would you interpret “Inertia and gravity are phenomena identical in nature.” This is much to clear to be a German/ English translation problem.

Well, that was not a clear way for Einstein to have said it. But the sense in which gravity is like a inertia has been explained already. Locally, there is no difference between being at rest in a gravitational field and being at "rest" in an accelerating reference frame. The statement that gravity is the same as inertia doesn't mean very much.

It's really more worth your time trying to understand General Relativity, rather than understanding specific things Einstein might have said about it. His words are not Holy Writ.

 

[stevendaryl]

Understood, but I think for some people the statement “the force you feel when you are at rest in a gravitational field is not 'the force of gravity'", can cause as much confusion as clarification.

Maybe so, but it's the kind of confusion that actually leads to a deeper understanding. You can't feel gravity--you only feel resistance to gravity. So you don't feel gravity pushing you down, but you feel the floor holding you up against gravity.

 

[A.T.]

Without getting too technical, we can largely equate tidal gravity with curvature. So it's common to say that Einstein's elevator has a uniform gravitational field (even though an object will weight slightly different amounts at the top and bottom of the elevator), and thus it has no tidal gravity and no curvature.

Also see this post by DrGreg for the difference between elevator frame (B) and intrinsic curvature with tidal forces (C):

This is my own non-animated way of looking at it:

• A. Two inertial particles, at rest relative to each other, in flat spacetime (i.e. no gravity), shown with inertial coordinates. Drawn as a red distance-time graph on a flat piece of paper with blue gridlines.
  
• B₁. The same particles in the same flat spacetime, but shown with non-inertial coordinates. Drawn as the same distance-time graph on an identical flat piece of paper except it has different gridlines.
  
  B₂. Take the flat piece of paper depicted in B₁, cut out the grid with some scissors, and wrap it round a cone. Nothing within the intrinsic geometry of the paper has changed by doing this, so B₂ shows exactly the same thing as B₁, just presented in a different way, showing how the red lines could be perceived as looking "curved" against a "straight" grid.
  
• C. Two free-falling particles, initially at rest relative to each other, in curved spacetime (i.e. with gravity), shown with non-inertial coordinates. This cannot be drawn to scale on a flat piece of paper; you have to draw it on a curved surface instead. Note how C looks rather similar to B₂. This is the equivalence principle in action: if you zoomed in very close to B₂ and C, you wouldn't notice any difference between them.

Note the diagrams above aren't entirely accurate because they are drawn with a locally-Euclidean geometry, when really they ought to be drawn with a locally-Lorentzian geometry. I've drawn it this way as an analogy to help visualise the concepts.

 

[A.T.]

What his equivalence principle says is that the force of gravity is the same (or is the same sort of force) as the "inertial forces" that you feel when accelerating.

Even this isn't quite right. A better statement would be that the force you feel when you are at rest in a gravitational field is the same sort of force as the inertial force you feel when accelerating. But the force you feel when you are at rest in a gravitational field is not "the force of gravity".

I find both formulations confusing, because both talk about "feeling inertial forces". Inertial forces can be used to explain coordinate acceleration in a non-inertial frame, but you can "feel" only the frame invariant proper acceleration from interaction forces.

I would put it like this:

Coordinate acceleration towards a mass (in coordinates where the mass is at rest) is attributed to:
- An interaction force by Newton
- An inertial force by Einstein

 

[MikeGomez]

Also see this post by DrGreg for the difference between elevator frame (B) and intrinsic curvature with tidal forces (C)...

"B2. Take the flat piece of paper depicted in B1, cut out the grid with some scissors, and wrap it round a cone. Nothing within the intrinsic geometry of the paper has changed by doing this, so B2 shows exactly the same thing as B1, just presented in a different way, showing how the red lines could be perceived as looking "curved" against a "straight" grid."

So there actually is curvature due to proper acceleration, correct? If so, it seems to me that would mean that in the case of Einstein’s elevator, there would curvature, even if the Weyl curvature tensor would (almost) completely vanish.