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

[Dr_Mike_J]

TL;DR Summary

Gravity and Force

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

 

[Dale]

Summary:: Gravity and Force

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

It doesn’t. The spring is stretched by the contact force at the top. If you remove the contact force then it will stop stretching.

 

[Dr_Mike_J]

It doesn’t. The spring is stretched by the contact force at the top. If you remove the contact force then it will stop stretching.

What is the origin of the contact force?

 

[sophiecentaur]

It doesn’t. The spring is stretched by the contact force at the top. If you remove the contact force then it will stop stretching.

The OP has to realize that your answer will need reading and re-reading, followed by a lot more reading round this topic. GR is actually VERY HARD and it can be quite a struggle to relate it to our experience. So was Newton's Physics in his time.

 

[Dale]

What is the origin of the contact force?

I don’t know. You didn’t describe the scenario completely. It could be a person’s hand or it could be glue or a screw or a weld. It doesn’t matter what the origin is.

 

[etotheipi]

Imagine you lay the massive spring horizontally on a smooth ice rink, and pull one end with a constant force. The spring stretches!

 

[sophiecentaur]

@Dr_Mike_J See what I mean! I foresee some brain ache on the horizon.

 

[Dr_Mike_J]

I don’t know. You didn’t describe the scenario completely. It could be a person’s hand or it could be glue or a screw or a weld. It doesn’t matter what the origin is.

OK, more detail.
Spring hanging from a hook on a ceiling.
A 1kg mass on the end of the spring held by a human hand such that the spring is at its natural length unstretched.
You let go of the mass slowly avoiding any adiabatic changes so that the spring smoothly attains its new equilibrium configuration so there is now a force acting through the whole spring F=kx.
My question is what is the origin of this force within the spring which is the sum of all the interatomic attractions within the metal of the spring caused by the displacement of the constituent atoms from their equilibrium positions.

 

[Dale]

My question is what is the origin of this force within the spring which is the sum of all the interatomic attractions within the metal of the spring caused by the displacement of the constituent atoms from their equilibrium positions.

The contact force from the hook on the top.

 

[sophiecentaur]

Give him a break!

 

[Dale]

What? That is the GR answer. In GR that is the only external force acting on the spring/mass system.

Edit: hmm I just realized he may not have intended the mass to be part of the spring, but rather separate. In which case it would be the pair of contact forces from the mass and the hook.

 

[sophiecentaur]

I told him there would be some brain ache.

 

[Dr_Mike_J]

My question is quite genuine. I think there is a real conceptual problem here.
Springs stretch because an external force pulls the constituent atoms away from each other.
If gravity is not a force how does the spring get deformed by a hanging mass.
GR says that it's the warp of space-time causing the mass to try to move along the new geodesic generated by the presence of the Earth. Electrostatic intermolecular theory says that the spring stretches when the constituent atoms are caused by an external force to increase their average separation against the force generated by their intermolecular potential. So where does this force originate if the gravity that balances it is not a force?

 

[Dale]

My question is quite genuine

Yes, I assume you are being genuine in your question, as am I in my answer.

If gravity is not a force how does the spring get deformed by a hanging mass.

Even in Newtonian mechanics gravity doesn’t deform a spring. It is always the contact forces which deform the spring. This is not unique to GR, although it may be that GR challenges you to think about it correctly by not labeling gravity as a force.

So where does this force originate if the gravity that balances it is not a force?

The contact force(s) on the spring.

Think about more scenarios (use Newtonian mechanics), some with gravity and some without, some at rest some accelerating, some with contact forces and some without. Identify which scenarios have the spring in tension and which do not. You will see that you can have a spring in tension with or without gravity, but that you must have the contact forces. Therefore the origin of the tension is the contact forces not gravity.

 

[Nugatory]

14 posts in and no one has linked @A.T.'s video yet?
Watch this video, imagine that the apple is connected to the branch by a spring... the spring will stretch until it is exerting a force sufficient to pull the apple back onto the "hang" worldline.

 

[ZapperZ]

We get this type of question A LOT, and we need to go back several paces here.

Just because something can be viewed another way, it doesn't mean that the first or original way of looking at it isn't valid anymore. Let's look at the concept of classical force in general. Notice I didn't say WHAT type of force, just a generic force.

If you look at classical mechanics, a force F can also be looked at as nothing more than a gradient of the potential energy field V, i.e.

F = - dV/dx

where I've restricted this to 1D for simplicity since I don't know if the grad operator is understood here.

Then, by the same logic, does it mean that ALL forces don't really exist, but instead it is simply a gradient in the potential field? How did you pull on the spring then?

So the question here is that, are you aware of this, and if you do, why aren't you asking the same question about all forces, be it gravity or electrostatic, etc.?

Zz.

 

[Ibix]

Actually, @etotheipi's comment about a mass and spring lying horizontally on ice is a nice example. The spring stretches if you pull one end, even though gravity is completely irrelevant here.

The point is that in GR, the curvature of spacetime is such that the unaccelerated path of the mass is to fall to the floor, just as the (horizontally) unaccelerated path of the mass on the ice is to just sit there. In either case you need to apply a force to get it to follow an accelerated path, and that force comes from whatever's holding the spring at the other end. It's this force that stretches the spring.

 

[DrStupid]

Springs stretch because an external force pulls the constituent atoms away from each other.
If gravity is not a force how does the spring get deformed by a hanging mass.

There is a similar effect in classical mechanics: Just take a spring with two masses at the ends and let it rotate around the common center of mass. If centrifugal force is not a force (in fact it isn't) how does the spring gets deformed by the rotating masses?

 

[Dr_Mike_J]

OK Before the mass is put on the end of the spring the spring is unstretched. When the mass is put on the end of the spring what causes the spring to stretch?

 

[DrStupid]

When the mass is put on the end of the spring what causes the spring to stretch?

The electromagnetic force between mass and spring.

 

[A.T.]

OK Before the mass is put on the end of the spring the spring is unstretched. When the mass is put on the end of the spring what causes the spring to stretch?

Note that causation is not relevant to force analysis.

 

[Ibix]

OK Before the mass is put on the end of the spring the spring is unstretched. When the mass is put on the end of the spring what causes the spring to stretch?

The spring starts pulling the mass out of its freefall trajectory with a proper acceleration of 1g. Just like the mass-on-ice example, the force at one end of the spring and the resistance to acceleration of the mass lead to the spring stretching.

 

[A.T.]

Even in Newtonian mechanics gravity doesn’t deform a spring. It is always the contact forces which deform the spring. This is not unique to GR, although it may be that GR challenges you to think about it correctly by not labeling gravity as a force.

I think the above is key here. A local uniform gravitational field cannot deform anything. Neither in Newtonian gravity nor in GR. If the gravity is not uniform, it will deform the spring, even without anything attached to it.

 

[pervect]

Let's try a different question. Suppose you are on an accelerating elevator, and you have the same hanging spring.

How would you describe the origin of the springs stretch in this case? What are the real forces on the hanging spring in the accelerating elevator?

Spoiler

The only real force on the spring is the force making applied to make it accelerate. Other forces are fictitious forces.

 

[Dr_Mike_J]

Going back to the spring hanging from the ceiling...
When the mass is on the point of being hung on the end of the spring, that is the surface of the material of the hook of the mass is in contact with the surface of the material of the spring there are balanced electrostatic repulsions and attractions between their constituent positive and negative charges - the spring is in equilibrium.
As the mass is allowed to hang from the spring what is the reason for the spring's loss of equilibrium?

 

[A.T.]

When the mass is on the point of being hung on the end of the spring, that is the surface of the material of the hook of the mass is in contact with the surface of the material of the spring there are balanced electrostatic repulsions and attractions between their constituent positive and negative charges - the spring is in equilibrium.

Sounds like you are confusing Newtons 3rd Law (equal but opposite forces between mass and spring) with Newtons 2nd Law (force equilibrium on the spring).

 

[DrStupid]

As the mass is allowed to hang from the spring what is the reason for the spring's loss of equilibrium?

The spring starts excerting a force on the mass, pulling it ouf of its geodesic. The corresponding counter force (together with the force from the ceiling that keeps the top of the spring in place) stretchs the spring. There are no other forces involved.

 

[Sagittarius A-Star]

Summary:: Gravity and Force
How then does it cause a spring to stretch?

You need a force to accelerate a mass, in an inertial frame. The hook is accelerating with 9.81 m/s² upwards, relative to the local inertial system in a nearby free falling elevator cabin. If you force the mass to get the same acceleration, a contact force is needed. You can regard the spring as an acceleratometer.

But be aware: If the spring stretches, you store mechanical energy in it. According to E=mc², it gets more mass by the stretching and the needed force is even greater. -:)

 

[Dale]

OK Before the mass is put on the end of the spring the spring is unstretched. When the mass is put on the end of the spring what causes the spring to stretch?

The contact force from the mass.

Did you go through the exercise I suggested? What scenarios did you consider and what did you find?

As the mass is allowed to hang from the spring what is the reason for the spring's loss of equilibrium?

The contact forces on the ends of the spring. They were zero before stretching and nonzero after stretching.

This is getting a little repetitive.

 

[Dale]

To break the repetitiveness let me try a different approach. I will propose many different scenarios including if the spring is stretched or unstretched. You identify the forces acting on the spring itself, and assume that the spring itself has mass. We are only interested in forces along the direction of the spring, not transverse forces.

1) spring is stretched horizontally attached to two walls, supported by a frictionless table to keep it straight

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

3) spring is attached to the ceiling and is stretched slightly under its own weight

4) spring is stretched between my hand and a mass while accelerating horizontally across a frictionless table

5) spring is unstretched lying on a table

6) spring is stretched between two masses in horizontal uniform circular motion on a frictionless table

 

[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.