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

[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.