Gravity & Springs: Exploring the Connection

[Sagittarius A-Star]

Nope, I don't think we really need the notion of a pseudo-force. But see my other comments.

Thank you for this clarification! Then, because of the principle of equivalence, we also don't need a pseudo-force in the reference frame of Einstein's accelerated elevator in flat spacetime.

When I am in this elevator and throw a ball, then I could calculate it's parabolic trajectory with "variational calculus" under the condition of the principle of maximum proper time and based on the pseudo-gravitational time-dilation in this frame (if I knew, how to do "variational calculus" ).

I believe I've seen that general approach used in "Exploring black holes". ((I could be mistaken, unfortunately, my memory isn't what it used to be)).

I will check this. Thank you very much for this hint!

However, it's also true that in popularizations, and in history, we also do talk about "gravity" on Einstein's elevator.

I think, there is a difference between "gravity" and"gravitation":

The gravity of Earth, denoted by g, is the net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation).

Source:
https://en.wikipedia.org/wiki/Gravity_of_Earth

I decided, to use the term "pseudo-gravitational time-dilation" in Einstein's elevator.

Einstein used the term "gravitation" also in flat spacetime and also wrote a justification for this:

But the distinction between “pseudo-gravity” and “true gravity” is precisely what Einstein denied. The equivalence principle asserts that these are intrinsically identical. Einstein’s point hasn't been fully appreciated by some subsequent writers of relativity textbooks. In a letter to his friend Max von Laue in 1950 he tried to explain:

...what characterizes the existence of a gravitational field from the empirical standpoint is the non-vanishing of the Γlik, not the non-vanishing of the [curvature]. If one does not think intuitively in such a way, one cannot grasp why something like a curvature should have anything at all to do with gravitation. In any case, no reasonable person would have hit upon such a thing. The key for the understanding of the equality of inertial and gravitational mass is missing.

Source:
https://www.mathpages.com/rr/s5-06/5-06.htm

 

[PeterDonis]

Suddenly the Earth disappears into nothingness

It can't; this would violate conservation of energy. It is pointless to propose scenarios that violate the laws of physics, and then ask what the laws of physics say about them.

 

[PeterDonis]

I think, there is a difference between "gravity" and"gravitation"

A better way of putting it would be that both terms, "gravity" and "gravitation", have been given multiple overlapping meanings in the literature, so it's better not to use them at all if you want precision, but to use more precise terms instead, like "coordinate acceleration in an inertial frame" or "tidal gravity" or "spacetime curvature".

 

[pervect]

Thank you for this clarification! Then, because of the principle of equivalence, we also don't need a pseudo-force in the reference frame of Einstein's accelerated elevator in flat spacetime.

If one has a Lagrangian, one doesn't need a force to determine the equations of motion. One can just solve the Euler-Lagrange equations, which can be derived from the variational methods you describe. Similarly, if one has a free particle and a metric, one can find the equations of motion of the free particle from the geodesic equation directly from the metric, by calculating the associated Christoffel symbols. The Christoffel symbols can be computed from sums of various partial derivatives of the metric coefficients.

Simply by writing down the Rindler metric for the accelerated elevator, one could write down the geodesic equations to determine the equations of motion of a free particle. See https://en.wikipedia.org/w/index.php?title=Geodesics_in_general_relativity&oldid=953775102. And for Christoffel symbols, see https://en.wikipedia.org/w/index.ph...7278134#Christoffel_symbols_of_the_first_kind. The Wiki entry on Rindler Coordinates currently even gives the geodesic equations, so there is no need to work it out.

Of course, the principle of covariance says that the motion of a free particle will be independent of the coordinate choice. Rindler coordinates are just a coordinate choice, the end result that the free particle in flat space-time undergoes inertial motion regardless of the coordinates used.

Typically, solving the geodesic equations directly is a bit messy, it is advantage of symmetries of the metric which give rise to "constants" or "integrals" of motion. The generators of these symmetries are called "Killing Vectors".

 

[Sagittarius A-Star]

I am sure there are many who would say that philosophy and logic are not irrelevant to physics.

Long ago, physics was a sub-topic of philosophy. My impresion is, that both subjects got then more and more separated. That became clear in a public discussion in 1922 in Paris between the philosopher Henri Bergson (who wrote several books about "time", also one about the "twin paradox") and the pysicist Albert Einstein. Bergson used a similar argument as yours:

Philosophy, he modestly argued, still had a place. Einstein disagreed. He fought against giving philosophy (and by inference Bergson) any role in matters of time.
…
Bergson temporarily had the last word during their meeting at Société française de philosophie. His intervention negatively affected Einstein’s Nobel Prize, which was given “for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect” and not for relativity.

Source:
https://dash.harvard.edu/bitstream/...nandtheexperimentthatfailed(2).pdf?sequence=2

 

[DrStupid]

What is the most recent received definition of "force"?

According to Newton's original definition a force is an external influence that compels a body to change its state of rest or uniform motion in a staight line. In order to adjust that to general relativity you need to generalise the force-free motion to geodesics: A force is an external influence that compels a body out of the geodesic. Gravity doesn't do that in general relativity but it defines the geodesics.

 

[sqljunkey]

If gravity is just curvature in spacetime, why does mass curve spacetime in the first place though.

 

[Ibix]

If gravity is just curvature in spacetime, why does mass curve spacetime in the first place though.

There is no answer to that at our current level of understanding. Quantum theories of gravity may help, but will presumably have different "but why..." questions underlying them.

 

[sqljunkey]

Ibix are you saying there may still be a particle that might be mediating the force of gravity? That the curvature of spacetime is an artifact of this mediating force field ?

 

[Sagittarius A-Star]

If gravity is just curvature in spacetime, why does mass curve spacetime in the first place though.

We don't know the "why", but we have experimental evidence, "that" mass curves spacetime, by observing "geodetic precession" (satellite experiment "Gavity Probe B" and observation of binary pulsars).
Source:
https://www.einstein-online.info/en/spotlight/geodetic_precession/#Geodetic_precession

 

[Ibix]

Ibix are you saying there may still be a particle that might be mediating the force of gravity? That the curvature of spacetime is an artifact of this mediating force field ?

I'm saying that a future theory of gravity (which we assume will be quantized) might provide an explanation for curvature and how it happens. This may or may not involve a mediating particle. My (limited) understanding is that some candidate theories do include a graviton, some don't. You'd probably do better searching this forum rather than hijacking this thread further.

 

[sqljunkey]

well then the good thing about GR is that it means General Theory of Relativity and not General Theory of Gravity.

 

[pervect]

If we define a philosophical question as something that can't be determined by experiment, it's not a topic that can possibly be addressed scientifically.

This doesn't necessarily make it unimportant, but because philosophical questions can't be answered by experiment, discussions of them tend to go on endlessly. See for instance Feynman's remarks in his essay "Is Electricity Fire". Science takes the view that it is more productive to work on questions that can actually be answered, via experiment.

This is the scientific philosophy. But, of course, one needs some philosophy to have a scientific philosphy :). I do personally believe that looking at the evidence is a good thing, and I often wish that people would do it more often. But it's not something one can convince people to do if they don't want to listen, sadly.

I do find, though, that sometimes a discussion of the underlying philosophy is helpful, perhaps even necessary, for people to understand science, to understand why a theory makes the predictions it does. This only works if the person has an open enough mind to listen, though - if they have some preconceived notions about their personal philosphy, sometimes they just won't listen, and as a result they don't understand why a theory makes the predictions it does, or in some cases argue that it doesn't actually make those predictions. Pointing to the literature where the predictions are made can be helpful sometimes, thpugh unfortunately, I can say (by experiment) not always.

 

[Dr_Mike_J]

Little did I realize the hornet's nest which would get stirred up by my initial question "How then does it (gravity) cause a spring to stretch? However I feel I have learned much form the various exchanges. It seems to me that things can be summed up by saying that the latest understanding of things is that gravity is not itself a force though it can lead to the occurrence of contact forces by the presence of mass (in this case the Earth) distorting space-time so that in the presence of sufficient mass in large enough concentration a spring with a modest mass on one end radially closer to the centre of the said mass away from a fixing point radially further away will stretch. It is contact forces which cause the spring to stretch but the magnitude of those contact forces is influenced by the space-time distortion caused by the presence of the large concentrated mass. At present we do not understand how the Earth distorts space-time but it may be that this distortion is mediated by particles called gravitions. Would it be that if sufficient evidence for gravitons could be found we would then be able to say that gravity is a force?

 

[Dale]

It is contact forces which cause the spring to stretch

Yes, exactly.

Would it be that if sufficient evidence for gravitons could be found we would then be able to say that gravity is a force?

In Newtonian physics gravity is a force, so I am content with calling gravity a force even today without such evidence.

 

[Sagittarius A-Star]

Galilei: "natural horizontal motion is motion at constant velocity, and natural vertical motion is falling at constant acceleration"
Newton: Free fall is caused by a vertical force.
Einstein: Free fall is the natural motion (similar to Galilei).
Quantum Gravity: TBD.

 

[Sagittarius A-Star]

by the presence of mass (in this case the Earth) distorting space-time so that in the presence of sufficient mass in large enough concentration a spring with a modest mass on one end radially closer to the centre of the said mass away from a fixing point radially further away will stretch.

At the top end of the spring, the "fixing point" (= "Hook") and the spring pull with equal forces at each other.
At the bottom end of the spring, the "modest mass" an the spring pull with equal forces at each other. The spring stretches because of pull forces at each end.
The contact force at the "modest mass" must not to be canceled by another force, because it has a proper acceleration (=not following it's geodesic) and F = m * a.

 

[A.T.]

Newton: Free fall is caused by a vertical force.
Einstein: Free fall is the natural motion (similar to Galilei).

I think the key is that Einstein redefines which frame is inertial and which is non-inertial.

Newton:
- Surface frame is inertial
- Downward acceleration in the surface frame is caused by an interaction force

Einstein:
- Surface frame is non-inertial
- Downward acceleration in the surface frame is caused by an inertial force

This gets more to the core of the difference than semantic discussions about what should be called a "force".

 

[sqljunkey]

GR predicts WAVES which were observed by LIGO. WAVES traveling in spacetime, probably interacting with each other, in physical world. Which to me at least says that GR's spacetime curvature has real physical meaning in real life. Not that I ever thought other wise...

 

[PeterDonis]

Which to me at least says that GR's spacetime curvature has real physical meaning in real life.

We already knew that long before LIGO. Spacetime curvature is tidal gravity. We have huge amounts of of evidence for tidal gravity.

 

[Sagittarius A-Star]

Einstein:
- Surface frame is non-inertial
- Downward acceleration in the surface frame is caused by an inertial force

Mentioning, which frame is inertial and which not, is very helpful!

But I think, speaking about an "inertial force" in context of this discussion about GR, although not wrong, creates irritation. The "inertial force" (="pseudo force") can be omitted in GR, according to my above discussion #71 with @pervect. It has no physical "added value". I prefer to use the arguments from my above posts
#86

Einstein: Free fall is the natural motion

and #87

The contact force at the "modest mass" must not to be canceled by another force, because it has a proper acceleration (=not following it's geodesic) and F = m * a.

see also:

In order to adjust that to general relativity you need to generalise the force-free motion to geodesics: A force is an external influence that compels a body out of the geodesic. Gravity doesn't do that in general relativity but it defines the geodesics.

 

[A.T.]

But I think, speaking about an "inertial force" in context of this discussion about GR, although not wrong, creates irritation. The "inertial force" (="pseudo force") can be omitted in GR, according to my above discussion #71 with @pervect. It has no physical "added value".

You can omit "inertial force" and model the non-inertial frame using curve-linear coordinates, like in the video:


But I think, if someone is already familiar with inertial vs- non-inertial frames in Newtonian mechanics, and thus also knows the difference between interaction and inertial forces, then this is a much simpler first step to take:

"Gravity is an inertial force now"

compared to:

"There is no force of gravity, and you have to understand differential geometry to explain why an apple falls."

 

[Sagittarius A-Star]

You can omit "inertial force" and model the non-inertial frame using curve-linear coordinates, like in the video:
...

Yes. But, as going to curve-linear coordinates does not change the reference frame, it must also be possible to calculate the (almost) parabolic trajectory in the normal coordinates in the accelerated frame without inertial forces. I described in my above posting #61 the approach by making use of the "principle of maximum proper time".

"Gravity is an inertial force now"

That would mean for example, that two forces pull at the Earth moving around the sun: One inertial force in the direction of the sun and another one (centrifugal force) in the opposite direction. That sounds too similar to the theory of Newton. I think it is easier to say, that the Earth follows with no force the ellipse in space, because this is locally a straight line in curved 4D-spacetime.

 

[A.T.]

That would mean for example, that two forces pull at the Earth moving around the sun: One inertial force in the direction of the sun and another one (centrifugal force) in the opposite direction.

The inertial force description only works locally, in approximately uniform fields. It's basically the Equivalence Principle, as a first step to transition from Newtonian mechanics.

I think it is easier to say, that the Earth follows with no force the ellipse in space, because this is locally a straight line in curved 4D-spacetime.

"Easier to say" is not always "easier to understand". At some point you have to go there, to explain the global picture, but the local Equivalence Principle is a good starting point.

 

[Dale]

it must also be possible to calculate the (almost) parabolic trajectory in the normal coordinates in the accelerated frame without inertial forces. I described in my above posting #61 the approach by making use of the "principle of maximum proper time".

This is just a question of semantics. When you do the calculations you will get a term that is proportional to the mass. You can derive it from GR and call it a Christoffel symbol. You can derive it from Newtonian gravity and call it a real force. You can derive it from the equivalence principle and call it an inertial force. You can derive it from a variational principle and not call it anything at all. Regardless of what you call it or how you derive it, the term is present in the equations of motion in those coordinates and not in other coordinates.

 

[sqljunkey]

Well I believe that because of gravitational waves, GR strips away the meaning of force. Newtonian gravity has 0 notions that waves exist. By saying there is a dynamic fabric that constantly interacts with mass and also with itself, (probably), GR is saying spacetime curvature is not a artifact of some external force field.

In theory , I don't know for sure, two waves can come and cancel each other out. Leaving zero force, from the way I understand it.

PeterDoris might be able to expand more on this though.

 

[A.T.]

...semantic discussions about what should be called a "force".

...strips away the meaning of force

There we go again...

 

[PeterDonis]

I believe that because of gravitational waves, GR strips away the meaning of force.

The fact that GR does not treat gravity as a force is not just due to gravitational waves. GR treats all aspects of gravity as due to spacetime geometry.

In theory , I don't know for sure, two waves can come and cancel each other out. Leaving zero force, from the way I understand it.

Your understanding is flawed.

Gravitational waves are not "waves of force". What Newtonian physics calls "gravitational force" is present in GR in cases where there are no gravitational waves present at all.

 

[Sagittarius A-Star]

When you do the calculations you will get a term that is proportional to the mass.

O.K. My understanding of this is: The OP described here only scenarios, which are corner-cases of GR (locally, weak gravitation, small velocities), in which alternatively GR, SR (in accelerated frame) or Newton's theory can be used. Therefore, you must get with all approches the same motion formula for free fall in the surface-reference frame, containing in vertical direction a term "m * a", with "a" being the coordinate-acceleration. In case of using SR, you may or may not call this term a pseudo-force.

 

[pervect]

Well I believe that because of gravitational waves, GR strips away the meaning of force. Newtonian gravity has 0 notions that waves exist. By saying there is a dynamic fabric that constantly interacts with mass and also with itself, (probably), GR is saying spacetime curvature is not a artifact of some external force field.

In theory , I don't know for sure, two waves can come and cancel each other out. Leaving zero force, from the way I understand it.

GR certainly says that an electric field exerts a force, so your statement that "GR strips away the meaning of force" isn't accurate. Perhaps you'd care to try again?

Going back to my general remarks about science making physical predictions, and philosophy being about things that can't be experimentally tested, let's look at a few experimental results.

1) When we measure "gravitational mass" and "inertial mass", they've always turned out the same. For instance, see wiki on "tests of the weak equivalence principle", https://en.wikipedia.org/w/index.ph...52193#Tests_of_the_weak_equivalence_principle, and note that current tests say that the two agree within a few parts in 10,000,000,000,000, i.e. a few parts in 10^13.

2) The Pound Rebka experiment, and other later experiments with atomic clocks, show that the rate of clocks on the Earth depends on their height in the Earth's gravity. This effect is so significant that it's taken into account when averaging the readings of atomic clocks to create our base time standards, including TAI which is the bases for Universal Coordinate Time.

These results are both predicted from GR, and I believe that the predictions were made before the experiments were done. Do you have any alternative explanation for these results? If so, what is it?

Lastly, but perhaps the most importantly: what experimental predictions, if any, does the notion that "gravity is a force" actually make? Follow up questions , applicable only if it does make predictions, are "what predictions are they" and "have they been tested".

I have a few additional comments about point 2, "gravitation's effects on time". There is no evidence that I'm aware of, that forces such as electric fields, would cause similar time dilation. Do you think there should be such an effect, if so, do you have some theoretical basis for it? Has anyone, before the existence of the effect was noticed, made such a prediction about electric fields? Note that GR predicted this result before the experiments were done.

If "gravity is just a force, like an electric field", shouldn't both forces cause similar effects on clocks?

As far as your wave ideas go, there is no precedent for colliding waves cancelling out. I suspect it's not possible, but I don't have a formal proof. It seems rather speculative, at least, to suggest that they should cancel and worse to then use that speculation as an argument against GR.

 

[sqljunkey]

well since I've been asked to try again, I'll try again. People in this thread have been saying that as soon as you have curvature in the spacetime you will see these test masses start free falling along these geodesics. Gravitational waves, are curves in spacetime. So I would assume that as soon as these waves hit test masses they will start free falling alongside those geodesics. And I extrapolated that since these waves move along one spacetime they will probably interact with one another. And there could be a hypothetical situation where these waves in spacetime cancel each other out and create a total flat space. Now there will probably be a geodesic this test mass free falls along. This could all be happening near a massive star.

In my opinion when they came with the idea of gravitational waves they were saying that GR was in fact the theory of gravity. It didn't matter anymore if there was no mass or no energy in the south, you can still be compelled to free fall in that direction.

 

[PeterDonis]

People in this thread have been saying that as soon as you have curvature in the spacetime you will see these test masses start free falling along these geodesics.

Test masses will free fall along geodesics whether spacetime is flat or curved. Curvature just changes the relationship between different geodesics.

Gravitational waves, are curves in spacetime. So I would assume that as soon as these waves hit test masses they will start free falling alongside those geodesics.

They won't "start" free falling--they'll be free falling the whole time. But as a result of the gravitational wave passing, the relationship between the different geodesics that nearby test masses are free-falling on will change. That is exactly what a gravitational wave detector like LIGO is detecting: the change in the relative motion of different test masses due to a gravitational wave passing.

I extrapolated that since these waves move along one spacetime they will probably interact with one another.

Since the Einstein Field Equation is nonlinear, yes, in principle gravitational waves can interact with one another. In practice, any gravitational waves we can detect here on Earth are so weak that the nonlinearities are negligible, and they can be treated like linear waves, which just superpose without any interaction between them--as, for example, EM waves do.

there could be a hypothetical situation where these waves in spacetime cancel each other out and create a total flat space

You should first ask yourself if this is possible with EM waves: can EM waves cancel each other out and create a region where there are zero EM fields whatsoever? And if so, how large can such a region be?

(Hint: the answer is that while EM waves can cancel each other out at particular points, they can't cancel each other out completely over an extended region.)

This could all be happening near a massive star.

Near a massive star, even at points where incoming gravitational waves do cancel each other out, that doesn't remove the static spacetime curvature due to the star. You can't cancel out that kind of spacetime curvature with gravitational waves at all.

In my opinion when they came with the idea of gravitational waves they were saying that GR was in fact the theory of gravity.

This is nonsense. GR was known to be a theory of gravity as soon as it was invented, and this had nothing to do with its prediction of gravitational waves. It had to do with all the predictions it made that matched Newtonian gravity, plus the additional predictions it made that accounted for phenomena that couldn't be explained by Newtonian gravity (like the precession of Mercury's perihelion), and also predicted new phenomena that had never been observed (like bending of light by the Sun and gravitational redshift).

It didn't matter anymore if there was no mass or no energy in the south, you can still be compelled to free fall in that direction.

I have no idea what you mean by this.

 

[pervect]

well since I've been asked to try again, I'll try again.

I noticed you didn't answer any of my questions, though. Did you think about them at all? They might help you understand a bit more about what GR does say and doesn't say.

People in this thread have been saying that as soon as you have curvature in the spacetime you will see these test masses start free falling along these geodesics.

Another poster has mentioned that test masses always free fall along geodesics, regardless of curvature, by definition, and has addressed some of the other thigns you have said fairly well.

 

[sqljunkey]

alright fine gravity is the curvature in spacetime.