B How Does Gravity Affect Spring Stretching in Einstein's Theory?

[PeterDonis]

Is this a coordinate-dependent definition?

No. A more technical way to state it is that a geodesic is a curve that parallel transports its own tangent vector along itself.

I think, a coordinate-independent definition would be to say, geodesics are world lines of objects not influenced by real forces.

If by "real forces" you mean "forces that cause proper acceleration", i.e., that are actually felt, then your definition is indeed coordinate-independent and equivalent to the definition I gave above (though showing the equivalence takes some work).

 

[PeterDonis]

I feel the downward radial force of their feet.

And they feel the upward force of your shoulders. So there is a force in both directions (Newton's Third Law).

Also, you feel an upward force from the ground, and the ground feels a downward force from you (again Newton's Third Law).

So focusing on downward forces is simply ignoring half of the forces present.

Thus I am experiencing the result of being in proximity to the earth.

No, that's not the cause. The ISS is almost as close to the center of the Earth as you are, and if you were standing on a tower whose height was the same as the orbital altitude of the ISS, and someone else stood on your shoulders, the force you felt (and the force they felt) would be virtually the same. So "proximity to Earth" can't be the cause.

I connect the two experiences and deduce that there is a downward radial pressure

No, that's not what you should deduce. As noted above, there are forces in both directions, downward and upward. So just looking at the downward forces can't be right.

Why can we not use the word "force"?

You can use the word "force" in GR for things that are felt as forces. Gravity is not felt as a force. Astronauts in the ISS are just as much subject to Earth's gravity as you are, yet they feel no force. So "gravity" cannot be the force you are feeling when you stand on Earth and someone else stands on your shoulders.

think we are in danger of losing sight of the fact that science is about making observations of the way the universe behaves and seeking to find patterns in these observations.

It is you who are losing sight of that, by ignoring the obvious pattern I just described in my previous paragraph above.

 

[Dale]

Excellent, thanks for taking the exercise seriously.

1) spring is stretched horizontally attached to two walls, supported by a frictionless table to keep it straight
There is a tension in the spring acting horizontally through its whole length;

OK, so in the lengthwise direction we have contact force from the wall, no gravity, and tension.

The spring is touching the table so there is electrostatic repulsion between their outer clouds of electrons acting between them along the whole length of the spring (if this were done in a free-fall lab or on ISS there would be less repulsion between them than on the Earth)

Yes. And we are ignoring the transverse forces. I did it that way so that we wouldn’t have to go to deep space to get rid of gravity, but all of those horizontal scenarios could be replaced by going to deep space to get rid of gravity.

2) spring is unattached and unstretched as it free falls vertically

So gravity is present, contact forces are absent, and it is unstretched

3) spring is attached to the ceiling and is stretched slightly under its own weight
Contact force at the attachment to the ceiling, small tension in the spring due to its weight

Excellent, so there is a small contact force, there is the usual gravity force, and there is a small tension.

4) spring is stretched between my hand and a mass while accelerating horizontally across a frictionless table
contact force between the spring and the hand causing the spring and mass to accelerate

Excellent, so there is a contact force, there is no gravity, and there is tension.

5) spring is unstretched lying on a table
see 1)

So no contact force, no gravity, and no tension.

6) spring is stretched between two masses in horizontal uniform circular motion on a frictionless table
The tension in the spring supplies the centripetal force needed to keep the two masses moving in a circle

Finally, we have contact forces, no gravity, and tension.

So in summary we can put the results in the following table:
$$\begin{array}{cccc}
\text{Scenario} & \text{Contact} & \text{Gravity} & \text{Stretching} \\
1 & \text{Yes} & \text{No} & \text{Yes} \\
2 & \text{No} & \text{Yes} & \text{No} \\
3 & \text{Yes} & \text{Yes} & \text{Yes} \\
4 & \text{Yes} & \text{No} & \text{Yes} \\
5 & \text{No} & \text{No} & \text{No} \\
6 & \text{Yes} & \text{No} & \text{Yes} \\
\end{array}$$
Looking at the summary it is pretty clear. Whenever there is a contact force there is stretching, regardless of whether or not there is gravity. Whenever there is no contact force there is no stretching, regardless of whether or not there is gravity. And in 4 we found a small contact force and a small amount of stretching, so the amount of stretching is even related to the amount of the contact force.

So clearly, gravity does not cause the stretching. Scenarios 1, 2, 4, and 6 all contradict the idea. Also, this is all done using Newtonian physics and treating gravity as a force. Even in that case it is clearly not gravity that causes a spring to stretch, it is the contact forces.

My question relates to how we describe the difference in interaction between, for example, my feet and the floor on Earth compared with my feet and the floor of the ISS were I to have the privilege of being there.
Am I not allowed to call my perception of a force a force?

Sure. Such forces are are also called inertial forces or fictitious forces (I prefer the first). But what you cannot do is claim that they cause a spring to stretch.

 

[Dale]

The ISS is almost as close to the center of the Earth as you are, and if you were standing on a tower whose height was the same as the orbital altitude of the ISS, and someone else stood on your shoulders, the force you felt (and the force they felt) would be virtually the same. So "proximity to Earth" can't be the cause.

This is a really good point. @Dr_Mike_J I would recommend thinking about this quite a bit. Gravity is still present in the ISS and is in fact almost the same as on the surface of the earth. So the absence of a force in the ISS is not attributable to a lack of the gravitational force.

 

[MikeeMiracle]

Feel the need to thank all the contributers to this thread. I learned about contact forces, special thanks to Dale and his questions method, made it very clear.

 

[Dr_Mike_J]

Thank you @Dale for your patience.
Your table does not constitute evidence that gravity has no part in the stretching.
Just because stretching of the spring happens in situations where there is no gravity does not mean that gravity does not cause stretching.
Consider taking the spring and the same mass to the moon where the strength of gravity is about 1/6 of that at the surface of Earth. You would find that the extension of the spring would be about 1/6 that found on Earth.
It's hard not to make the conclusion that gravity has something to do with the stretching of the spring.
The contact forces don't arise out of nowhere they come from the system altering its position variables so as to reach equilibrium. So what is the mechanism by which this adjustment to attain balance comes about?
Do not forces have to be balanced for equilibrium or is that a purely Newtonian concept?

 

[PeterDonis]

Your table does not constitute evidence that gravity has no part in the stretching.

Gravity as a force has no part in the stretching. That's what we've been talking about.

Gravity as spacetime curvature does have a part in the stretching, since it determines what worldlines the individual pieces of the spring would follow if there were no internal forces in the spring and no contact forces between the spring and masses or suspension points. Which in turn determines what those forces have to be to make the individual pieces of the spring follow the worldlines they actually follow.

Consider taking the spring and the same mass to the moon where the strength of gravity is about 1/6 of that at the surface of Earth. You would find that the extension of the spring would be about 1/6 that found on Earth.

Yes, but this is because of gravity as spacetime curvature, not gravity as a force. The spacetime curvature in the vicinity of the Moon is different than in the vicinity of the Earth.

 

[Dale]

Just because stretching of the spring happens in situations where there is no gravity does not mean that gravity does not cause stretching.

Actually it does. Scenarios 1, 4, and 6 show that gravity is not necessary for stretching. Scenario 2 shows that gravity is not sufficient for stretching. In contrast the contact forces are both necessary and sufficient. So it does very clearly identify the causality.

Consider taking the spring and the same mass to the moon where the strength of gravity is about 1/6 of that at the surface of Earth. You would find that the extension of the spring would be about 1/6 that found on Earth.

Sure, but you would also find that the contact force is about 1/6 of that found on earth. So this scenario also supports the claim that the contact force is the cause of the stretching. That was the point of scenarios 3 and 5. Even when it seems like gravity is the explanation, that is just because it is going along with the contact force.

To distinguish which is the cause you must vary them separately, not together. This is a common mistake in experimental design.

Do not forces have to be balanced for equilibrium or is that a purely Newtonian concept?

They do, but equilibrium is a frame dependent concept. In an inertial frame the spring attached to the ceiling is not in equilibrium. It is accelerating.

But in any case equilibrium is different from stretching. So something could be necessary for equilibrium but not for stretching and vice versa.

 

[A.T.]

Do not forces have to be balanced for equilibrium or is that a purely Newtonian concept?

Do you actually understand Newtonian mechanics? In particular, the difference between inertial and non-inertial frames of reference? Or the difference between interaction and inertial forces? Or the difference between Newtons 2nd and 3rd Law?

I would strongly suggest getting a good grasp on these concepts before starting with GR.

 

[pervect]

Can you please describe the deeper connection?

Do you have a link to a description of this deeper connection between forces and time dilation?

I don't think, that you need a concept with a fictitious force at Einstein's elevator in space. In the accelerated elevator, you have a pseudo-gravitational time-dilation. If a lamp at the ceiling of the elevator sends a light-pulse, it will be received blue-shifted by a sensor at the floor of the elevator. From the viewpoint of an external inertial observer, the reason is the Doppler effect. This pseudo-gravitational time-dilation curves gedesics. The concept does not differ locally from the "real" gravitation of the earth.

Actually, I rather think we do. Consider any other sort of force. Say for instance, an electric field. Do electric fields, no matter how strong cause time dilation, for an observer at rest? The answer is basically no, with the exception that a strong enough electric field could cause gravitational effects, which would then cause time dilation. But we already know that gravitational effects case time dilation, the point of the argument is to consider what happens in the non-gravitational case. And in the non-gravitational case, forces do not cause time dilation.

So, if forces do not cause time dilation, but gravity does, something different is going on with gravity - it's not "just a force". What is it then? Basically, we can regard "gravitational force" as the pseudo-force associated with describing things in a non-inertial frame. And the other effects, such as the effects on time, are related to our choice of using an accelerated frame.

In Newtonian mechanics, the only effect of going to an accelerated frame of reference is to introduce a "fictitious force", equal and opposite to the acceleration of the observer. A special relativistic analysis of the accelerated observer shows that things are much more complicated.

See for instance the Wiki article on "Rindler Coordinates", https://en.wikipedia.org/w/index.php?title=Rindler_coordinates&oldid=962334733

Rindler coordinates are basically the special relativistic equivalent of the Newtonian "accelerated frame". A few highlights. We can see the height-dependent time dilation in the form of the associated metric, namely

$$ds^2 = -\alpha^2 x^2 dt^2 + dx^2 + dy^2 + dz^2$$

The space-time geometry in these coordinates can be regarded as a normal spatial geometry, but the description of time in this geometry is different - the relationship between proper time, $\tau$, that a clock measures, and the coordinate time, t, imposed by our coordinate system, depends on the coordinate height, x.

The "deeper connection" that I mention is simply the fact that time and space are unified into a 4-dimensional space-time. The origin of this unification are not in GR, but in SR. The best exposition of this unification is "The Parable of the Surveyor", in Taylor & Wheeler's "Space-time physics.

To recap my argument. Trying to shoe-horn gravity into the mold of a force fails in many ways, one of the most obvious and basic is that forces cannot cause time dilation, while gravity can. Regarding gravity as a pseudo-force , associated with an accelerated frame of reference, rather than as a force, is a step forwards in understanding gravity, though it is far from complete. While it is far from complete, regarding it as a pseudo-force allows one to retain some of the intuition from Newtonian mechanics. While it is not complete, it's a step forwards. People used to Newtonian mechanics know that in Newtonian mechanics, the only effect of going to an accelerated frame is to introduce a pseudo-force. They tend to assume things are the same in special realtivity - they are not the same. The effects of an accelerated frame are more complicated, as the analysis in Wiki shows, and these effects involve time as well as space. This may seem mysterious to someone with a strictly Newtonian viewpoint, but when one understands why time and space are unified (a topic for another thread, I think), it becomes less mysterious.

There is a reason that we talk about the geometry of space-time as best describing General relativity, rather than talking about gravity being a "force". It's not just word salad.

 

[Sagittarius A-Star]

The "deeper connection" that I mention is simply the fact that time and space are unified into a 4-dimensional space-time. The origin of this unification are not in GR, but in SR. The best exposition of this unification is "The Parable of the Surveyor", in Taylor & Wheeler's "Space-time physics.

The "The Parable of the Surveyor" seems to explain the invariant spacetime-interval:
http://spiff.rit.edu/classes/phys200/lectures/intro/parable.html

To recap my argument. Trying to shoe-horn gravity into the mold of a force fails in many ways, one of the most obvious and basic is that forces cannot cause time dilation, while gravity can.

Agreed. Gravity is not a real force.

While it is far from complete, regarding it as a pseudo-force allows one to retain some of the intuition from Newtonian mechanics.

My question is, if we really need pseudo-force and an intuition from Newtonian mechanics.

Reason of my question is, that I found a really good video "General Relativity: Principle of maximum proper time" from Professor Josef Gaßner. Unfortunately, it is in German. I think you can understand the mathematical formulas he is writing and I can write a short English summary:

First he derives the principal of maximum proper time from the classical principal, that the integral about the Lagrange function is a minimum. His derived formula for proper time contains the sum of a velocity-dependent part and a gravity potential dependent part.

Then he throws an orange, that follows a parabolic trajectory. He explains this trajectory step by step with the appoach of the orange, to accumulate maximum proper time. The orange tries for example to stay longer in high altitude, where it can acquire more proper time, but not too high, because that would need a too high velocity for a too long time, which would reduce the gain of proper time.

The parabolic trajectory can be calculated from the principal of maximum proper time, I think by "variational calculus". A "Gravity force" is not needed to calculate the parabolic trajectory. Professor Josef Gaßner did not mention "pseudo-force". Do we really need it?

It seems, according to Peter Donis, that I used formally the wrong formulation "curved geodesic", when I meant a parabolic trajectory.

Here is the video (unfortunately in German), sorry:

 

[PeterDonis]

It seems, according to Peter Donis, that I used formally the wrong formulation "curved geodesic", when I meant a parabolic trajectory.

Yes, the parabolic trajectory is a geodesic (actually, it's a uniform field approximation to the true geodesic, which will be a segment of an elliptical orbit about the Earth's center). The geodesic looks curved when plotted in space, but it is straight in spacetime (or as straight as a curve can get in the curved spacetime around the Earth).

 

[DrStupid]

OK someone is standing on my shoulders. I feel the downward radial force of their feet. I know from experience that this radial force depends on the mass of the person above me. Thus I am experiencing the result of being in proximity to the earth.

... or the result of being in an accelerating rocket or in a centrifuge.

I also know by people's accounts that this experiment performed on the ISS results in very little force experienced.

That seems you accept that there is very little gravitational force in a space station. What do you think keeps this guy on the ground:

So I connect the two experiences and deduce that there is a downward radial pressure which seems compatible with the notion of a force.

... or of a fictitious force.

Why can we not use the word "force"?

Because this thread is in "Special and General Relativity" and not in "Classical Physics".

 

[pervect]

The "The Parable of the Surveyor" seems to explain the invariant spacetime-interval:
http://spiff.rit.edu/classes/phys200/lectures/intro/parable.html

Yes.

Agreed. Gravity is not a real force.My question is, if we really need pseudo-force and an intuition from Newtonian mechanics.

We don't necessarily need it, IMO - but it's handy. Especialy if we're talking to someone who thinks gravity is or should be a force. We can then say "Well, in part it's a pseudo-force", but that's not the whole story, so they can build on their outlook. But certainly there are other approaches.

Reason of my question is, that I found a really good video "General Relativity: Principle of maximum proper time" from Professor Josef Gaßner. Unfortunately, it is in German. I think you can understand the mathematical formulas he is writing and I can write a short English summary:

First he derives the principal of maximum proper time from the classical principal, that the integral about the Lagrange function is a minimum. His derived formula for proper time contains the sum of a velocity-dependent part and a gravity potential dependent part.

Then he throws an orange, that follows a parabolic trajectory. He explains this trajectory step by step with the appoach of the orange, to accumulate maximum proper time. The orange tries for example to stay longer in high altitude, where it can acquire more proper time, but not too high, because that would need a too high velocity for a too long time, which would reduce the gain of proper time.

The parabolic trajectory can be calculated from the principal of maximum proper time, I think by "variational calculus". A "Gravity force" is not needed to calculate the parabolic trajectory. Professor Josef Gaßner did not mention "pseudo-force". Do we really need it?

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

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)). The principle of maximum proper time has various other names, one of which is "the principle of maximal aging". It's good for calculating trajectories of objects if you know the metric.

It looks like there have a second edition of "Exploring Black Holes" that's only published online . It can be found at Taylor's website, http://www.eftaylor.com/exploringblackholes/. But I haven't read it.

The principle of maximal aging is not quite sufficient for understanding Einstein's equations fully, IMO, though, as it doesn't give you much insight into how the metric is determined from the matter content. If we use Wheeler's adage, "Spacetime tells matter how to move, matter tells spacetime how to curve", the principle of maximal aging answers the first question, it explains how spacetime tells matter how to move, but it doesn't really demonstrate how "matter tells space-time how to curve"

The second part is explained by Einstein's field equations, $G_{uv} = 8 \pi T_{uv}$ in geometric units, there's some additional factors in non-geometric units.

One of my favorite explanations of the second part is Baez & Bunn's "The Meaning of Einstein's Equation". You can find it on arxiv, also at Baez's webstie . http://math.ucr.edu/home/baez/einstein/einstein.pdf

We promised to state Einstein's equation in plain English, but have not done so yet. Here it is

Given a small ball of freely falling test particles initially at rest with respect to each other,the rate at which it begins to shrink is proportional to its volume times the energy density at the center of the ball, plus the pressure in the x direction at that point, plus the pressure in the y direction, plus the pressure in the z direction.

However, in the footnotes they mention

To see why equation (2) is equivalent to the usual formulation of Einstein's equation, we need a bit of tensor calculus.

It turns out there is actually quite a bit of tensor calculus, and it involves knowing how some entities transform to go from their statement about a ball of test particles to the usual and full formulatio of Einstein's equations. To my mind there are actually extra unstated assumptions hidden in the footnotes. But I still like the paper, it gave me a lot of insight.

I am also fond of regarding gravity not as a force, but as the curvature of space-time as given by the Riemann curvature tensor. I also like to decompose this tensor via the Bell decomposition into various parts, but that's another story, though I will mention that one of these parts is just tidal gravity in Newtonian physics.

While this is very useful, and how I look at gravity, and also has reasonable support in the literature (MTW does this, I think), it can be confusing. I just got through talking at length about "gravity" on Einstein's elevator. With the Riemann tensor approach, I would probably be arguing instead that there is no gravity on Einstein's elevator, because the space-time there is flat.

And it's true that the space-time on the elevator is flat, and it's true that we often talk about gravity as curved space-time. However, it's also true that in popularizations, and in history, we also do talk about "gravity" on Einstein's elevator.

So, there are different way of looking at things, and while the math in the end is perfectly self-consistent, English language descriptions in general may not be.

 

[Nugatory]

So, there are different way of looking at things, and while the math in the end is perfectly self-consistent, English language descriptions in general may not be.

The truth is indeed in the math... but I have often thought that the English language explanations would be better if we all were in the habit of using “gravitational acceleration” to refer to what we observe in Einstein’s elevator, and “tidal gravity” to refer to the effects of spacetime curvature.

 

[Dr_Mike_J]

Indulge an ageing juvenile and let's consider a gedanken experiment.
I am sitting on a chair which rests on the Earth's surface and enjoying the security that sensing the contact forces give me. Suddenly the Earth disappears into nothingness in the same way that Douglas Adams's whale appeared. It would appear that the contact forces between myself and the chair would become vanishingly small. It appears that the contact forces were only there in the former situation because the Earth was there and that it may justifiably be said that the Earth caused the contact forces, that is, the phenomenon of gravity causes the forces. The exact cause can be enquired into and at present the best theory we have is Einsein's theory of General Relativity. However that does not remove the implication that gravity causes forces to come into being. I suppose the problems we have are to do with the inherent inadequacy of language. What is the most recent received definition of "force"?

 

[Dale]

It appears that the contact forces were only there in the former situation because the Earth was there and that it may justifiably be said that the Earth caused the contact forces, that is, the phenomenon of gravity causes the forces

A few points.

First: even if gravity causes forces that does not imply that gravity is a force itself. Causes of forces need not be forces themselves.

Second: spring stretching is not a force, so even if gravity causes forces that does not imply that gravity causes spring stretching.

Third: an analysis of other scenarios will show that gravity is neither a necessary nor a sufficient condition for the contact forces between the chair and your butt. As such, it is not the cause of those forces.

Fourth: you came here with a question which has been clearly answered. You no longer seem to be trying to learn about general relativity. You now seem to be pushing a private viewpoint. Please focus on learning rather than arguing.

 

[A.T.]

I suppose the problems we have are to do with the inherent inadequacy of language.

Your problem seems twofold:

- You complain about aspects of GR that equally apply to Newtonian mechanics, because you apparently don't understand Newtonian mechanics in the first place.

- You are focusing on issues that are irrelevant to physics and mainly philosophical or semantic.

 

[Dr_Mike_J]

@A.T. I am sorry you feel that way. I do object to assumptions regarding what I do or do not understand.
I am sure there are many who would say that philosophy and logic are not irrelevant to physics.
@Dale I thank you again for your patience. I do not agree that my initial question has been answered but I am happy to agree to disagree. Thank you for what I found was, for the most part, a really stimulating discussion.

 

[Dale]

I do not agree that my initial question has been answered

How can you disagree?

Q:

According to Einsteins general theory of relativity gravity is not a force.
How then does it cause a spring to stretch?

A: gravity does not cause a spring to stretch, even in Newtonian mechanics.

I am sorry, but to disagree that your question has been answered is factually false. The fact that it has been answered is clearly observable.

What I see is that you came with a question, and received some answers that challenged your preconceived notions (about Newtonian gravity among other things), but you do not want to adjust your thinking. But how can you hope to learn if you are not willing to change your preconceptions?

I am happy to agree to disagree

I am willing to agree to disagree regarding the correctness of the answers, but not their existence. It is an observable fact that your question has been answered, even if you reject those answers. Saying that your question has not been answered is dismissive of the effort that respondents put into the answers. Dismissing other’s efforts is not a good way to participate in a community, particularly not a community of experts where respondents have put in decades of effort merely to be in a position to answer questions like yours.

 

[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.