# I Is Gravity inertia, acceleration or curvature in GR?

1. Mar 11, 2016

### fog37

Hello Forum,

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

2. Mar 11, 2016

### Staff: Mentor

Where did you read that? It is not.
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.

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.

3. Mar 11, 2016

### fog37

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

4. Mar 11, 2016

### 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?

5. Mar 11, 2016

### A.T.

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

Relativity distinguishes inertial and non-inertial frames as well.

6. Mar 11, 2016

### Staff: Mentor

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).
Motion is always relative to a reference frame, independent of its acceleration.
We do not. We just note that we are accelerating. The car seat pushes you, therefore you accelerate.

7. Mar 11, 2016

### alw34

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

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.

8. Mar 11, 2016

### 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.

9. Mar 11, 2016

### alw34

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.

10. Mar 11, 2016

### Staff: Mentor

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

Last edited: Mar 11, 2016
11. Mar 11, 2016

### alw34

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.....

12. Mar 11, 2016

### MikeGomez

Einstein said it many times.

Exactly.

13. Mar 11, 2016

### Staff: Mentor

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.

14. Mar 12, 2016

### MikeGomez

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 Einstein

From 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

15. Mar 12, 2016

### Staff: Mentor

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

16. Mar 12, 2016

### alw34

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:

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.

17. Mar 12, 2016

### Staff: Mentor

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".

18. Mar 12, 2016

### alw34

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

19. Mar 12, 2016

### stevendaryl

Staff Emeritus
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.

20. Mar 12, 2016

### Ibix

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.

21. Mar 12, 2016

### Staff: Mentor

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".

22. Mar 12, 2016

### stevendaryl

Staff Emeritus
Yes, that's better.

23. Mar 12, 2016

### MikeGomez

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

24. Mar 12, 2016

### MikeGomez

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.
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.

25. Mar 12, 2016

### Staff: Mentor

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