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

[pervect]

A few more comments. In a previous post, I suggested that it was best to treat gravity as a fictitious force, rather than a real force. This is a rather subtle point, involving Einstein's equivalence principle, and could be argued. But "Einstein's elevator" is routinely used to explain General Relativity, henceforth GR, and why GR only has one sort of mass rather than two separate sorts of masses in Newton's theory. The two sorts of masses in Newton's theory are gravitational mass and inertial mass, and Newton's theory provides no explanation of why they should always be the same, while Einstein's theory assumes they are the same.

When we look closely at Einstein's elevator, we see that the force stretching the spring is actually not a real force, but a fictitious force. The real force is applied to one end of the spring. The other end of the spring applies a force to accelerate the mass.

This is good as far as it goes, but it doesn't go far enough, in my opinion. Many of the more interesting predictions of GR simply do not have a natural interpretation as a "force". One example of this is gravitational time dilation. Forces and time dilation are two different things, that are apparently unrelated. But there is a deeper connection, which can be seen by understanding the geometric version of the theory.

Another example of such a phenomenon that doesn't have a natural interpretation as a force are the predictions of GR of the changes in spatial geometry that occur near large masses (with a suitable definition of terms, most notably a definition of spatial geometry that involves splitting a static space-time geometry into a static spatial geometry and some notion of time).

Basically, attempting to unnaturally force all the predictions of GR into the paradigm of a "force" is not natural to the theory, and will lead to an incomplete understanding of the predictions of the theory. Some aspects of gravity are more amenable than others to a 'force' interpretation, but many interesting aspects of the theory are best explained from the geometrical viewpoint.

Fully understanding the geometrical viewpoint of GR is no easy task. But it has its rewards if one follows through. It is possible to understand some aspects of GR without understanding it's geometrical viewpoint, but generally such understandings are limited and incomplete.

 

[PeterDonis]

When we look closely at Einstein's elevator, we see that the force stretching the spring is actually not a real force, but a fictitious force. The real force is applied to one end of the spring. The other end of the spring applies a force to accelerate the mass.

The force applied by the spring to the mass is not fictitious, it's real. So both the forces you describe are real. What stretches the spring is that the second force is time delayed with respect to the first, because it takes time for the effects of the first force to propagate through the spring's material from one end to the other.

 

[Sagittarius A-Star]

When we look closely at Einstein's elevator, we see that the force stretching the spring is actually not a real force, but a fictitious force. The real force is applied to one end of the spring. The other end of the spring applies a force to accelerate the mass.

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.

 

[PeterDonis]

This pseudo-gravitational time-dilation curves gedesics.

You can't curve geodesics; geodesics are by definition straight.

The worldlines of objects at rest in the accelerated elevator are curved, but those worldlines are not geodesics. The time dilation effect observed in the elevator is a consequence, not a cause, of the fact that the worldlines are curved.

 

[pervect]

The force applied by the spring to the mass is not fictitious, it's real. So both the forces you describe are real. What stretches the spring is that the second force is time delayed with respect to the first, because it takes time for the effects of the first force to propagate through the spring's material from one end to the other.

I'd agree that if we view the spring-mass system in an inertial frame, where Newton's laws apply without any modifications, the force applied by the spring to the mass is real, and that it causes the mass to accelerate. And there is another force applied to the other end of the spring as well, and the spring is under tension, and this tension is "real".

In an accelerating frame with Newtonian physics, we need to modify Newton's laws to introduce the concept of a "fictitious force". The popularized argument based on Einstein's elevator equates this fictitious force in the accelerated frame to "gravity". So in this accelerated frame, the fictitious force we introduce is equated to gravity.

Without introducing significantly more mathematics (for instance Christoffel symbols), I don't see any better way of explaining things than to suggest that many (but not all) aspects of gravity are approximated by treating it as a fictitious, or inertial, force. The motivation for treating it this way arises form the equivalence principle. If gravitational and inertial masses are to be always equal, it is always possible to say that any gravitational force must be equivalent to some inertial force.

I don't think it's possible to do better than this without introducing some advanced mathematics such as Christoffel symbols. Also, as I mentioned in a subsequent post, I don't think the force paradigm is sufficient to capture all of the effects that GR predicts. Thus, people who insist on viewing gravity as a force because it's been done that way in Newtonian physics tend to miss out on and be confused by some of the more interesting aspects of GR that do not fit this restricted worldview.

 

[Dale]

When we look closely at Einstein's elevator, we see that the force stretching the spring is actually not a real force, but a fictitious force.

Fictitious forces cannot stretch springs. Only real forces can do that.

Consider, in an inertial frame there are no fictitious forces so the stretching of the spring is determined by the real forces. Changing to a non-inertial frame does not remove any real forces, so the explanation of the stretching in terms of real forces is unchanged. So the fictitious forces do not contribute to the stretching.

The fictitious forces are only needed to explain the motion of the object in the non-inertial frame. They do not explain the stretching or any other physical effect.

 

[PeterDonis]

In an accelerating frame with Newtonian physics, we need to modify Newton's laws to introduce the concept of a "fictitious force".

But this fictitious force is not what stretches the spring. As @Dale says, only real forces can do that.

 

[Sagittarius A-Star]

Forces and time dilation are two different things, that are apparently unrelated. But there is a deeper connection, which can be seen by understanding the geometric version of the theory.

Can you please describe the deeper connection?

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

 

[Dr_Mike_J]

Let's have a look at those spring scenario force analyses...

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;
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) the greater part of this repulsion will be due to the mass of the spring;
2) spring is unattached and unstretched as it free falls vertically
For free fall the spring will be accelerating due to the close proximity of the Earth (in GR does the existence of acceleration not necessarily imply a force? genuine question - my neurons are rusty!)
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 usually ignored in mechanical calculations - the proverbial "massless spring"!
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
5) spring is unstretched lying on a table
see 1)
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 plus
electrostatic repulsion between their outer clouds of electrons acting between them along the whole length of the spring and between the masses and the table
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?

 

[A.T.]

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?

The contact force between your feet and the floor is an interaction force (EM-repulsion).

 

[Dr_Mike_J]

So what causes the EM-repulsion to be greater when I'm standing on the floor on Earth compared with standing on the floor of the International Space Station?

 

[A.T.]

So what causes the EM-repulsion to be greater when I'm standing on the floor on Earth compared with standing on the floor of the International Space Station?

The EM-forces are distance dependent: If your atoms get too close to the floor atoms, they repel each other.

 

[Dr_Mike_J]

So, what causes the atoms of my feet to be closer to the floor atoms on Earth than to the floor atoms on the ISS?

 

[A.T.]

So, what causes the atoms of my feet to be closer to the floor atoms on Earth than to the floor atoms on the ISS?

The "cause" of the proximity is irrelevant for the EM-force. In general, questions about "the cause" are vague, because it's not clear what qualifies as a "cause". They are usually irrelevant to the actual physics (quantitative predictions).

In GR a "cause" that could be named here, is that everything tends to move along a geodesic in space-time (free fall). Your atoms are trying to free fall, while the floor atoms are not in free fall, and thus they get closer together, resulting in repulsion.

 

[Dr_Mike_J]

A.T. you seem to have an aversion to the notion of cause and effect.
The universe is replete with one thing causing another thing ("The fundamental interconnectedness of all things" [Dirk Gently], the Butterfly Effect etc).
Also what about the floor itself, why are the atoms of my feet trying to free fall whilst the floor atoms are not?

 

[PeterDonis]

Am I not allowed to call my perception of a force a force?

You never perceive gravity as a force; any force you actually feel will not be gravity, but something else. When you're standing on the floor on Earth, the force you feel is the floor pushing up on you, not gravity.

Your atoms are trying to free fall, while the floor atoms are not in free fall

The floor atoms are "trying" to fall just as your atoms are. See below.

what about the floor itself, why are the atoms of my feet trying to free fall whilst the floor atoms are not?

The floor atoms are trying to fall, but they are being pushed on by the atoms below them, and so on all the way down to the center of the Earth.

 

[Sagittarius A-Star]

geodesics are by definition straight.

Is this a coordinate-dependent definition? Does it depend on describing the geodesic in the coordinates of an inertial reference frame?

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

 

[A.T.]

Also what about the floor itself, why are the atoms of my feet trying to free fall whilst the floor atoms are not?

As I said, everything tends to move along a geodesic in space-time (free fall). The floor is prevented from free fall by some other upwards forces, just like you are prevented from free fall by the upwards force from the floor.

 

[Dr_Mike_J]

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. I also know by people's accounts that this experiment performed on the ISS results in very little force experienced. So I connect the two experiences and deduce that there is a downward radial pressure which seems compatible with the notion of a force. Why can we not use the word "force"? Does a force have to be communicated by billions of virtual particles? I 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. If we detach our theories from our own minds it is like Michelangelo burning the scaffolding he's lying on.

 

[A.T.]

...seems compatible with the notion of a force.

That's why it is modeled as an interaction force in Newtonian gravity. But being proportional to mass it can also be modeled as an inertial force, as is the case locally in GR.

 

[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.