# Is Newton's third law of motion consistent with GTR?

1. Mar 17, 2016

### Ahan Sha

If an apple hanging in the tree has only reaction upwards, then what will happen to a Newtons third law? how is it that there is no "force " downwards, but have spacetime curvature which "mimics" a force. why can't spacetime curvature be itselt a force?

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2. Mar 18, 2016

### Orodruin

Staff Emeritus
First of all, you seem to have a common misunderstanding of Newton's third law. It does not say that the force from the branch on the apple is the reaction force to gravity. The third law partner of the force from the branch on the apple is the force from the apple on the branch. Both of those forces are still there in GR.

Last edited: Mar 21, 2016
3. Mar 18, 2016

### A.T.

As Orodruin wrote, Newtons 3rd still applies to the forces between apple and branch in GR. But it doesn't apply to gravity acting on the apple, because in GR that is modeled as an inertial force. Inertial forces are not subject to Newtons's 3rd Law, even in classical mechanics.

An alternative to using inertial forces is using deformed coordinates:

4. Mar 21, 2016

### Ahan Sha

just
but... my doubt is still unclear. just forget about the branch. Only think about our Apple hanging on that. I thought, Apple is in equilibrium because of gravitational force downwards and reaction force upwards( according to Newtonian Gravity). but you said "It does not say that the force from the branch on the apple is the reaction force to gravity" if this is true, then how is Apple in equilibrium according Newtonian gravity) pls reply

5. Mar 21, 2016

### Orodruin

Staff Emeritus
In Newnonian mechanics, the force on the apple from the branch is equal to the gravitational force on the apple. However, this is just the equilibrium condition. It does not make the forces a third-law pair. The third law partner of the gravitational force on the apple from the Earth is the gravitational force of the Earth on the apple.

6. Mar 21, 2016

### Staff: Mentor

I was under the impression that gravity is just as subject to Newton's 3rd Law as any other force is.

7. Mar 21, 2016

### Orodruin

Staff Emeritus
He is talking about inertial forces. Gravity is not an inertial force in Newtonian mechanics.

8. Mar 21, 2016

### rcgldr

What about a 2 body system, two objects orbiting about a common center of mass? With Newtonian mechanics, each body exerts a gravitational force on the other, in what could be considered a Newton third law pair. How does GR describe this situation?

9. Mar 21, 2016

### Staff: Mentor

The answer to that is kind of complicated. It has to do with how GR describes gravity as the result of spacetime curvature and how objects react to this curvature. Try this link: http://curious.astro.cornell.edu/ph...force-how-does-it-accelerate-objects-advanced

10. Mar 21, 2016

### Staff: Mentor

Roger.

11. Mar 21, 2016

### Orodruin

Staff Emeritus
Each body is undergoing geodesic motion in space-time. The presence of energy, momentum, and stress affects the shape of space-time and therefore the motion of the bodies.

12. Mar 21, 2016

### rcgldr

I forgot to ask how GR affects rotating frames of reference. Assume a frame of reference that rotates at the same rate as a 2 body system with a circular orbit (so that the rate of rotation is constant). What is the effect on fictitious centrifugal and coriolis forces?

13. Mar 21, 2016

### Orodruin

Staff Emeritus
This is just putting a different set of coordinates on the same space-time.

14. Mar 21, 2016

### Staff: Mentor

15. Mar 21, 2016

### Staff: Mentor

Whether the Apple is in equilibrium or not is a frame-dependent concept.

In the frame of the Apple an accelerometer at rest measures an acceleration of 1 g upwards, so this frame is a non inertial frame with an inertial force of mg downwards. That inertial force downwards balances the upwards contact force from the branch and the Apple is at equilibrium.

In a local free-fall frame an accelerometer reads 0, so that frame is inertial and there are no inertial forces. The only force on the Apple is the contact force from the branch which accelerates the Apple upwards. So the Apple is not in equilibrium in that frame.

16. Mar 21, 2016

### haushofer

That depends on your notion of "inertial force"; I'd say it is :P

17. Mar 21, 2016

### pixel

From the Newtonian point of view, the earth is exerting a gravitational force, F1, downward on the apple and, from the 3rd law, the apple exerts an equal and opposite force, F2, upward on the earth. Also, the apple is exerting a force, F3, downward on the branch and, from the 3rd law, the branch is exerting an equal and opposite force, F4, upward on the apple. Now let's focus on the apple. Because it is in equilibrium, |F4| = |F1| i.e. they are numerically equal other than a sign. But they are not the forces to which the 3rd law applies.

18. Mar 21, 2016

### A.T.

Which notion of "inertial force" includes Newtonian Gravity?

19. Mar 21, 2016

### pervect

Staff Emeritus
Suppose one is in EInstein's elevator. Which is one of the usual pedagogical approaches to explaining how gravity works in GR. Would one say that "Newton's third law" still works in the elevator?

Personally, I would not, though I'm not sure how well defined the question is. What I would say is that none of Newton's laws apply directly in an accelerated frame of reference such as the elevator. It's not that the laws are wrong, it's just that they need to be applied in the proper manner, and the proper manner is to apply them in an inertial frame of reference. I would also say that inertial forces are not generally regarded as being real forces, so that if one is standing in an elevator, there is only one real force, that is the force on one's feet pushing you up to make you accelerate along with the elevator.

Now, I recall plenty of lectures (from as far back as high school physics) that "inertial forces are not real forces" but do not have any references handy on the issue. My understanding and recollection though, is that the reasoning is that real forces must transform as tensors, and inertial forces do not. It turns out that inertial forces transform as Christoffel symbols. Of course this isn't the understanding I had in high school, that understanding has developed over time. I believe that the difference between Christoffel symbols and forces is more apparent in generalized coordinates, but some differences are still there even in Cartesian coordinates.. For instance, a tensor that is zero in one coordinate system is zero in all coordinate systems - but inertial "forces" are present only in accelerated coordinate systems and not present in non-accelerated ones.

Having noted the distinction between inertial forces (represented by rank 1 tensors) and non-inertial forces (represented by non-tensorial Christoffel symbols), one logically concludes that gravity is a force in Newtonian mechanics, and not a force (but a non-inertial force) in GR.

The issue I see for B and I level students is making clear the distinction between inertial forces and non-inertial forces. I believe I'm recalling all this correctly, but it's been so long it's hard to be sure, and I am not sure where to check to make sure I haven't forgotten something important or added something that's a bit non-standard. There's also the issue that my explanation above used tensors, which may not be suitable for the target audience :(.

Perhaps the clearest route (and it's not as clear as I'd like) is to follow in Einstein's footstep, and go from the acceleating elevator in Newtonian mechanics to the accelerating elevator in special relativity. In Newtonian mechanics, we pay lip service to the difference between inertial forces and real forces, but the matter didn't seem terribly urgent at the time it was being taught. When we move on to special relativity, though, the differences become much more apparent, and the reasons for all those earlier, not terribly-well understood at the time cautions, becomes more plain in hindsight.

Consider the issue of time dilation. It's well known (though I'm not aware of any really basic non-tensor treatment) that if you have a pair of clocks in an accelerating elevator, they do not stay synchronized. Why does this happen? If inertial forces were actually forces, it wouldn't happen, as forces do not cause time dilation. But the coordinate transformation to an accelerated frame of reference (formally involving the Christoffel symbols) has properties that simply cannot be modelled by a force. So we're back to saying that the fundamental issue is to avoid conflating inertial forces and real forces. They may appear similar in Newtonian mechanics, but when one moves on to even special relativity, the difference between the two concepts becomes more apparent, and one will become confused if one does not make the proper distinctions between inertial forces and real forces.

And the corollary to all this is that while gravity is a real force in Newtonian physics, it's an inertial force in General Relativity.

20. Mar 21, 2016

### Staff: Mentor

They do if you are willing to include "inertial forces" in your analysis. See below.

No, there is also another real force, the force of your feet pushing back down on the floor of the elevator. As others have pointed out, this force is the one that is paired with the force of the elevator on your field by Newton's Third Law. So Newton's Third Law applies just fine.

Newton's Second Law also applies just fine in the non-inertial elevator frame, provided, as I said above, that you are willing to include an "inertial force" in this frame, which points downward (i.e., towards the elevator floor) and has a magnitude equal to the acceleration of the elevator times the mass of whatever object is being accelerated by it. So, for example, if you are standing on the floor of the elevator and let go a rock, the rock will be acted on by this inertial force, which causes it to accelerate downward. And the magnitude of this force obeys Newton's Second Law, $F = ma$.

Furthermore, if we want to explain why you, standing on the floor of the elevator, don't accelerate upward as a result of the force the floor exerts on you, we have to appeal to this same inertial force, which acts downward and has a magnitude exactly equal to that of the upward force of the floor on you. So the net force on you is zero, and you remain at rest in this non-inertial frame.

Of course some will object that all of the above is true only because we defined the "inertial force" in order to make it true. This is correct; but it doesn't make the above analysis invalid. It just makes it conceptually limited; we obviously can't use the above analysis to argue that inertial forces "must be real" (or words to that effect), because that would be arguing in a circle.

I think you mean this the other way around, correct?