# Einstein's Equivalence Principle.

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• FallenApple
In summary, the conversation discusses the concept of the equivalence principle in both classical Newtonian gravity and general relativity. It explains that in both situations, there is no way to tell the difference between being in an elevator in outer space or resting on Earth. However, the big insight in general relativity is that it explains the equality of inertial mass and passive gravitational mass, as well as the idea that gravity can be seen as an inertial force due to the geometry of space and time. This is in contrast to Newtonian gravity, where the equality of these masses is considered a coincidence and is not explained by the theory. The conversation also touches on the philosophical differences between the two theories and how general relativity goes beyond Newtonian gravity by
FallenApple
So if someone is in an elevator moving up at g in outer space, vs someone in an elevator in a uniform gravitational field of g, there is no way telling whether one is in an elevator in outer space or resting on earth.

I saw this clip where Brain Greene poked a few holes into a water bottle and dropped it, showing that in mid air, the water stops leaking, so the water is in free fall along with the bottle. But this "experiment" could have easily have been done in Newton's time. Surely Newton must have known. So where is the philosophical difference?
Is it because Einstein deduced that light must bend in an accelerating elevator? So the principle was already there, he just applied it in a novel way?

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FallenApple said:
There may be others, but I see two big insights.

The first is that it explains the equality of inertial mass and passive gravitational mass. It was known that they were equal, but their equality seemed a strange coincidence. This insight explained that they are equal because they are the same thing.

The second is that it means that gravity can be geometrized. Instead of being considered a real force it can be turned into an inertial force due to the geometry of space and time.
FallenApple said:
Isn't this true classically as well?
...
Surely Newton must have known.
There is a theory of gravitation, called Newton Cartan theory, that is a reformulation of Newtonian gravity in terms of the geometry of space and time. It was developed in the 1920's or 1930's I think, which was well after Newton. He did not know nor did anyone else, it just wasn't an avenue that had been pursued.

FallenApple, vanhees71 and Battlemage!
FallenApple said:
there is no way telling whether one is in an elevator in outer space or resting on earth.

Locally, that is, in a small enough patch of spacetime, no, there isn't.

FallenApple said:
Isn't this true classically as well?

It's also true in Newtonian gravity for ordinary non-relativistic objects, yes. Whether it's also true in Newtonian gravity for, say, light is debatable.

Also, the way Newtonian gravity deals, theoretically, with this experimental fact is different from the way GR deals with it. In Newtonian gravity, in order to predict that the equivalence principle will be obeyed, you have to assume that inertial mass and gravitational mass are equal, i.e., that the ##m## that appears in ##F = ma## is the same as the ##m## that appears in ##F = GmM/r^2##. There is no way to get that from the axioms of Newtonian physics; it has to be put in as an additional assumption. But in GR, the equality of inertial and gravitational mass is not even a question, because there aren't two different "masses" in the theory to begin with: the motion of an object in a gravitational field is a matter of spacetime geometry and has to be the same for all objects.

FallenApple, vanhees71, Battlemage! and 1 other person
One could imagine a material whose inertial mass was ##m##, but whose gravitational mass was ##km##. Newtonian physics handles such a material just fine - ##F=GKMkm/r^2=mg##. It's just a mysterious empirical fact that ##k=K=1## for all materials. Relativity treats gravity as geometry. Two particles released with the same velocity in the same place must follow the same path because otherwise the path isn't determined by geometry. Therefore, as Peter says, the equivalence principle is completely integrated into GR, while Newtonian gravity simply notes it as a curious coincidence.

It's not a mystery but the definition of the units of inertial and gravitational mass. That one is proportional to the other has to be put as an observable fact in Newtonian mechanics.

In General Relativity, mass is no longer source of the gravitational field, but it's the energy-momentum-stress tensor of matter (i.e., all kinds of non-gravitational fields), which is universally coupled to the gravitational field in the sense of a gauge theory (see the recent Insight article by @haushofer about this formulation of GR). In this sense GR goes way beyond Newtonian gravity.

Dale
FallenApple said:
Isn't this true classically as well?
It is true empirically, so any valid model must predict it.
FallenApple said:
So where is the philosophical difference?
This animation explains the difference in the interpretation of the observation:

There is also the fact that the (strong) POE states that the two situations are indistinguishable for all physics. There is no such expectation in Newtonian physics.

vanhees71 said:
It's not a mystery but the definition of the units of inertial and gravitational mass. That one is proportional to the other has to be put as an observable fact in Newtonian mechanics.
Indeed - you can fix k to have any value you like by a unit choice for gravitational mass. However, Newton gives no particular reason why it must have the same value for every material. But "k must be equal for everything" falls out of the Newtonian limit of GR because test particles move on geodesics, which means their motion can't depend on what they're made of.

vanhees71
Dale said:
There may be others, but I see two big insights.

The first is that it explains the equality of inertial mass and passive gravitational mass. It was known that they were equal, but their equality seemed a strange coincidence. This insight explained that they are equal because they are the same thing.

The second is that it means that gravity can be geometrized. Instead of being considered a real force it can be turned into an inertial force due to the geometry of space and time.

Oh ok. So if the equality of inertial mass and passive gravitational mass is not just coincidence, then it hints at the equality of both elevator situations.

By inertial force, you mean fictitious?

FallenApple said:
By inertial force, you mean fictitious?
Yes. I prefer the term "inertial", but "fictitious" is commonly used also.

PeterDonis said:
Locally, that is, in a small enough patch of spacetime, no, there isn't.
It's also true in Newtonian gravity for ordinary non-relativistic objects, yes. Whether it's also true in Newtonian gravity for, say, light is debatable.

Also, the way Newtonian gravity deals, theoretically, with this experimental fact is different from the way GR deals with it. In Newtonian gravity, in order to predict that the equivalence principle will be obeyed, you have to assume that inertial mass and gravitational mass are equal, i.e., that the ##m## that appears in ##F = ma## is the same as the ##m## that appears in ##F = GmM/r^2##. There is no way to get that from the axioms of Newtonian physics; it has to be put in as an additional assumption. But in GR, the equality of inertial and gravitational mass is not even a question, because there aren't two different "masses" in the theory to begin with: the motion of an object in a gravitational field is a matter of spacetime geometry and has to be the same for all objects.

Hmm interesting. I'm curious why light would behave differently in elevator 1 vs elevator 2 in the Newtonian scheme. I mean, in the Newtonian scheme, there is no experiment that can measure the difference, therefore according to science, there is no difference. Unless, under the Newtonian scheme, they define the inertial frame as one in which light does not bend.

[QUOTE="FallenApple, post: 5652767, member: 585259"I'm curious why light would behave differently in elevator 1 vs elevator 2 in the Newtonian scheme.[/quote]In Newtonian physics, massive objects in a gravitational field accelerate because they are being subjected to the force of gravity. However, light isn't a massive object, so there's no reason in the theory why light should be deflected by gravitational (or any other) forces at all and no reason to expect the behvior of light in the two elevators to be the same. We've already introduced the ad hoc assumption that gravitational mass is proportional inertial mass; but to explain the behavior of light in elevator #2 Newtonian physics needs another ad hoc assumption, namely that gravity acts on light as well.

## 1. What is Einstein's Equivalence Principle?

Einstein's Equivalence Principle is a fundamental concept in physics that states that the effects of gravity and acceleration are indistinguishable. This means that an observer in a closed laboratory cannot determine whether the laboratory is accelerating or in a gravitational field.

## 2. How did Einstein come up with the Equivalence Principle?

Einstein developed the Equivalence Principle as part of his theory of general relativity. He was inspired by the idea that an observer in a closed elevator would not be able to tell if they were experiencing the force of gravity or being accelerated in free space. This led him to conclude that gravity and acceleration must be equivalent.

## 3. What are the implications of the Equivalence Principle?

The Equivalence Principle has several important implications in physics. It is the basis for Einstein's theory of general relativity, which describes how gravity works on a large scale. It also helps us understand that gravity is not a force, but rather a result of the curvature of spacetime.

## 4. Are there any exceptions to the Equivalence Principle?

While the Equivalence Principle holds true in most cases, there are some exceptions. For example, at very small scales, such as the size of an atom, the effects of gravity can be distinguished from those of acceleration. Additionally, the Equivalence Principle does not apply in situations involving extreme gravitational fields, such as near a black hole.

## 5. How does the Equivalence Principle impact our understanding of the universe?

The Equivalence Principle has had a significant impact on our understanding of the universe. It has allowed us to make accurate predictions about the behavior of objects in gravitational fields, and has been confirmed by numerous experiments. It has also led to the development of new technologies, such as GPS, which rely on understanding the effects of gravity on time and space.

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