# Newton's 3rd Law and the Relativistic addition of velocities

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Does the latter follow from the former?

If two bodies  are in the same inertial frame of reference  and an attempt is made to  make them move wrt each other by physical measures limited to those 2 bodies  does it not follow that any increase in  velocity multiplied by mass  of one body must be  counterbalanced  an equivalent velocity multiplied by mass of the other body?

Does this  show that the increase in relative velocity cannot be linear and also that there must be a finite velocity for any two bodies which is a function of their combined masses?

Looked at another way ,in the famous example of a moving passenger on a moving train as viewed from the embankment, the moving passenger has to push the train in the opposite direction to his or her direction of movement...and so that shows you cannot just add velocities "on on top of the other"  from the pov of the observer on the embankment.

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The situation you describe is not subject to the velocity addition formula. I can observe something going 0.8c in the +x direction, and at 0.7c going in the -x direction, and note that their separation speed is 1.5c. That does not violate relativity.

Either of those obhjects would not see that. The speed of one relative to the other would require the formula to be applied.

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3 hours ago, swansont said:

The situation you describe is not subject to the velocity addition formula. I can observe something going 0.8c in the +x direction, and at 0.7c going in the -x direction, and note that their separation speed is 1.5c. That does not violate relativity.

Either of those obhjects would not see that. The speed of one relative to the other would require the formula to be applied.

Weĺl ,I was suggesting that the motion of the passenger on the train was caused entirely  by physical contact with the train ,so that their respective motions were linked .

This contact causes the train to move in the opposite direction depending on the relative size of their masses.

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7 hours ago, geordief said:

Does the latter follow from the former?

I think the former is the logical result of the latter.

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23 minutes ago, J.C.MacSwell said:

I think the former is the logical result of the latter.

Maybe it is just that I was taught the former before I learned the latter that Newton's 3rd law is second nature to me whereas  the relativistic addition of velocities seems more  complicated and in need of explaining)

Maybe that is why I  put the causal relationship  in that order (If indeed there is such a relationship between the 2 ideas)

Why do you think it might be the other way around ? (the "logical result" as you put it)

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4 minutes ago, geordief said:

Maybe it is just that I was taught the former before I learned the latter that Newton's 3rd law is second nature to me whereas  the relativistic addition of velocities seems more  complicated and in need of explaining)

Maybe that is why I  put the causal relationship  in that order (If indeed there is such a relationship between the 2 ideas)

Why do you think it might be the other way around ? (the "logical result" as you put it)

Newton's Laws of motion are generally considered low speed approximations of SR, and if you include Newton's Law of gravity that of GR.

As a more accurate version of reality SR/GR are well represented by Newton's Laws and much simpler for most applications.

However, though approximately consistent with it for most applications, I don't see how SR/GR would follow from Newton's Laws.

More accurate measurements where Newton's Laws were insufficient made SR/GR necessary to explain the discrepancies.

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Posted (edited)
8 hours ago, J.C.MacSwell said:

However, though approximately consistent with it for most applications, I don't see how SR/GR would follow from Newton's Laws.

Is it valid to ask if just this addition of relativistic velocities might be a "logical result" of Newton's 3rd Law**?

My thought process is based on my assumption that all relative motion is fundamentally  the result of physical interactions between objects ,groups of objects or bodies. (so only on the macro level).

That is why I asked for only 2 objects' relative motion to be considered (albeit with an observer in the same frame  of motion as both bodies at the outset)

(In any case of misunderstanding I am not suggesting one goes back to Newton's model in cases where Relativity is better or equivalent)

** that particular 3rd law has not been superceded in any way by General Relativity ,has it ?

Edited by geordief

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1 hour ago, geordief said:

Is it valid to ask if just this addition of relativistic velocities might be a "logical result" of Newton's 3rd Law**?

I don't think it's the logical result of Newton's 3rd Law alone, but certainly in part.

1 hour ago, geordief said:

** that particular 3rd law has not been superceded in any way by General Relativity ,has it ?

It's consistent with it, and not just as an approximation, so I would agree it's not superseded.

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2 hours ago, geordief said:

** ﻿that particular 3rd law has not been superceded in any way by﻿ General Relativity ﻿,h﻿as it ﻿?

No it hasn't.

As they are usually expounded Newton's 3 laws do not concur with Relativity.
In particular they do not meet the 'principle of relativity' that physical laws should have the same form in all (inertial) coordinate systems.

This treatment by Turner as a reformulation, starting with the 3rd Law, may interest some.

Note the comment, at the bottom of page 16,

"The only real Physics is in the third law,..."

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Thanks, Studiot! +1

I like it: the fact that the first 2 laws are in fact definitions.

You should have posted this a few years ago (-1), when I was heavily attacked by suggesting that at the bottom of Newton's laws is a circularity, and one of the reasons is that the laws partially are definitions.

OK, +1, and take an Advocaat from me.

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11 hours ago, geordief said:

Maybe it is just that I was taught the former before I learned the latter that Newton's 3rd law is second nature to me whereas  the relativistic addition of velocities seems more  complicated and in need of explaining)

Again, it doesn't really come into play. If the velocity addition formulae were a consequence of the third law, I would think that someone would have noticed before Einstein published stuff on relativity. As it is, it does seem in need of explaining, as it's a consequence of an invariant c, and that's not something that manifests itself in everyday phenomena that we observe with the naked eye.

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Posted (edited)

If we take my setup of two massive objects (in isolation save for an observer) being initially in the same frame of reference and proceeding to  increase their spatial separate  by means of converting their mass to energy ,then is it not undeniable that there exists a maximum separation velocity for  the system and that the increase in velocity is not linear?

If we increase the initial  size of the system does that maximum separation velocity approach a new limit  which will be equal to c ( for massive objects)?

So does Newton's 3rd law also imply/require c( a maximum finite speed)?

My next question would be ,if this 3rd Law does imply (or require) a value such as c, does it also require that this value be the same regardless of  frame of reference ( invariance is  the term,I think)

Edited by geordief

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20 minutes ago, geordief said:

If we take my setup of two massive objects (in isolation save for an observer) being initially in the same frame of reference and proceeding to  increase their spatial separate  by means of converting their mass to energy ,then is it not undeniable that there exists a maximum separation velocity for  the system and that the increase in velocity is not linear?

Even with a Galilean system, the increase in velocity is not linear. The speed will depend on the energy. It can be determined without invoking the velocity addition formula.

20 minutes ago, geordief said:

If we increase the initial  size of the system does that maximum separation velocity approach a new limit  which will be equal to c ( for massive objects)?

Yes, under relativity. In a Galilean system, there is no limit.

20 minutes ago, geordief said:

So does Newton's 3rd law also imply/require c( a maximum finite speed)?

By itself, no.

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(bold by me)

44 minutes ago, geordief said:

If we take my setup of two massive objects (in isolation save for an observer) being initially in the same frame of reference and proceeding to  increase their spatial separate  by means of converting their mass to energy ,then is it not undeniable that there exists a maximum separation velocity for  the system and that the increase in velocity is not linear?

This is interesting so I'll try to comment! Newton, used in isolation with no theory of relativity, has no upper limit for speed. But converting mass to energy is as far as I know not part* of Newton and implies that relativity is used? And then it is undeniable that there is a maximum speed c. But not due to Newton but due to the fact that relativity is introduced in the reasoning.

*) If "E=mc2" is used. Burning rocket fuel for instance I do not count as mass conversion

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4 minutes ago, Ghideon said:

(bold by me)

This is interesting so I'll try to comment! Newton, used in isolation with no theory of relativity, has no upper limit for speed. But converting mass to energy is as far as I know not part* of Newton and implies that relativity is used? And then it is undeniable that there is a maximum speed c. But not due to Newton but due to the fact that relativity is introduced in the reasoning.

*) If "E=mc2" is used. Burning rocket fuel for instance I do not count as mass conversion

Well I was imagining that all energetic resources could be used until there was no mass in the objects left. I see that Newton would probably not conceived of this but I  didn't see that as an obstacle.(I was imagining the objects being peopled by rocket engineers converting madly everything to fuel until there was absolutely nothing left)

I am happy though with Swansont's  last reply ,even if it has disappointed me. I have to bow to his greater knowledge and assume he is right if he is not contradicted by anyone else.

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40 minutes ago, geordief said:

Well I was imagining that all energetic resources could be used until there was no mass in the objects left. I see that Newton would probably not conceived of this but I  didn't see that as an obstacle.(I was imagining the objects being peopled by rocket engineers converting madly everything to fuel until there was absolutely nothing left)

I am happy though with Swansont's  last reply ,even if it has disappointed me. I have to bow to his greater knowledge and assume he is right if he is not contradicted by anyone else.

If you use all the mass (e.g. you annihilate matter and antimatter), all you can have left are massless particles which must travel at c.

If you use less, i.e. there is a massive payload, the speed limit for both payload and exhaust is c because relativity restricts you. No further analysis is necessary to know that this will be the asymptote

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19 minutes ago, geordief said:

I was imagining the objects being peopled by rocket engineers converting madly everything to fuel until there was absolutely nothing left

OT: I would recommend the engineers to save some oxygen and the hull of the rocket  ! Your description reminds me of when mr Fogg purchases the ship he is traveling with and burns the wooden parts of the ship as fuel since they have run out of coal. (Around the world in 80 days)

But back on topic. @swansont is correct. The math is very clear AFAIK, there is no upper speed limit in Newton. My comment below is more from a philosophical point of view.

I think you may have a point, sort of, with the rocket analogy. If we would discuss the issue with Newton maybe he would agree that there is an upper limit to the possible relative speed, but more from a practical/engineering point of view. With finite resources that are depleted* there could logically, from Newtons point of view, be some finite maximum speed for that specific situation. But that speed, according to Newton, could be much larger (or smaller) that speed of light. And the limit would be different from case to case and probably not derived from Newtons equations. In the formulas there would be no trace of an upper limit such as speed of light or similar, acceleration would be allowed to go on forever.

*) I'm not trying argue what Newton could have known or understood about various concepts that did not exist in his era.

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