Does the four-force, in some ways, invalidate the results of the three-force in SRT?

Question

I have recently submitted an article to PNAS, however, the editor, oddly enough, claimed that he has issues with the traditional Lorentz transformation for force, he stated:

The argument of this paper is based entirely on eqn. (3), for which the only references given are refs. (1) and (3). I am unable to access ref. (1) (and I suspect so will be most readers). As to ref. (3), eqn. (3) indeed corresponds to eqn. (12) of that reference, which in turn is derived from eqn. (7). However, no justification of the latter equation is given, and I believe that it is wrong, at least as regards the y-and z-components. It is true that there seems to be some ambiguity in the literature as to the correct definition of force in special relativity, but in the present context it seems to me that a satisfactory definition is (partial) rate of change of 4-momentum with respect to proper time. If that is correct then it transforms as the space component of a 4-vector, and the outcome is that the transverse force constants are unchanged in the moving frame, invalidating the author's eqn. (3) and with it his whole argument. It is possible that I am suffering from a blind spot here and that eqn. (3) is correct, but if so the author needs to provide explicit justification for it. Until that is done the manuscript has no claim to be considered for PNAS.

Eq. (3) in the article relates the spring constant in the rest frame to that measured by an inertial observer moving perpendicular to the spring alignment, which somehow shows that the transverse force conponents are measured smaller. Ref. [3] of the article belongs to O. Gron (Covariant formulation of Hooke’s law) in which my Eq. (3) is proved. The editor, however, tried to invalidate Gron's article, and when I resubmitted my article with more clarification, he said:

... I said that any resubmission which claimed without explicit demonstration that in SR the transverse component of force is subject to Lorentz contraction would be automatically rejected ... As regards ref. [5]- the fact that a particular textbook (which because of Covid-19 conditions I am unable to consult) may use a different definition of force from what I would regard as the proper one in SR (rate of change of relativistic momentum with proper time) is not a matter of sufficient interest as to justify a PNAS publication. No doubt there may be real issues to be sorted out regarding the concept of force in SR, but the present manuscript makes no contribution to the debate.

Ref. [5] was indeed Resnick's book of Introduction to Special Relativity. Do we have different force transformations in SRT or did the editor just try to get rid of the article by this absurd excuse?

I specifically want to know if the use of four-force would, in some ways, invalidate the fact that the transverse component of force is subject to Lorentz contraction as we know from the three-force transformation. It seems that the editor thinks so according to the boldfaced sentence.

Answer 1

That is probably referring to the standard definition of the four-force:

https://en.m.wikipedia.org/wiki/Four-force

The four-force is the derivative of the four-momentum with respect to proper time. Since the four-momentum is covariant and since the proper time is a Lorentz scalar, the four-force is manifestly covariant. It is commonly regarded as the correct relativistic generalization of the Newtonian concept of force.

Answer 2

This is a meaningless argument. There are different ways of setting up relativistic dynamics. You can write your equations either in terms of the three-force $d\mathbf{p}/dt$ or the four-force $dp^\mu / d\tau$. One approach doesn’t invalidate the other, because both will give the exact same results if applied consistently.

It is also not true that one definition is manifestly superior to the other. For example, the Lorentz force in terms of electric and magnetic field vectors is naturally a three-force. But the Lorentz force in terms of the electromagnetic field tensor is naturally a four-force. Both can be useful depending on the situation. (On the other hand, the choices $d\mathbf{p}/d\tau$ and $dp^\mu/dt$ are probably unambiguously bad; I've never seen them be good for anything.)

Of course, if you somehow think you’re written a paper that has proved that special relativity is not Lorentz invariant, that probably is a result of misusing the three-force. The editor’s request for you to switch to four-forces is then a completely reasonable one, as it will quickly expose the error.

Answer 3

knzhou and Dale are both correct. Here is how to express it in standard notation.

1. Four-force $$ F^a \equiv \frac{dP^a}{d\tau} $$ where $P^a$ is 4-momentum and $\tau$ is proper time.

2. Three-force $$ {\bf f} \equiv \frac{d{\rm p}}{dt} $$ where ${\rm p}$ is three-momentum and $t$ is time in the chosen inertial frame in which the three-force is being calculated. For electromagnetic force, for example, we have $$ {\bf f} = q ( {\bf E} + {\bf v} \times {\bf B} ) $$ In this equation $q$ is an invariant scalar and everything else is a frame-dependent $3$-vector.

Note, this 3-force is not the spatial part of the 4-force. The spatial part of the 4-force is $$ F^i = \gamma f^i $$ This is like the relationship between 4-velocity and 3-velocity.

Now one can of course also consider the quantity $d {\bf p}/d\tau$. It is a perfectly well-defined quantity but it is not necessarily the "preferred" definition of 3-force in special relativity as your referee thought. It may be preferred by some but it is not preferred by all and indeed I think there may not be a majority in favour of it. I think it is a bit of a monster and not much use. There are various ways to see this. One good way is to note the Lorentz force which I already mentioned: it is much more natural in terms of $\bf f$. Another interesting point is that in the case of the 3-force as I have defined it one finds, for a rest-mass-preserving force, the nice result $$ \frac{ dE}{dt} = {\bf f} \cdot {\bf v} $$ where here $E$ is total energy of the particle or system being pushed by the three-force ${\bf f}$ and $\bf v$ is its velocity. This leads to the notion of rate of doing work, and a connection between force and energy (in addition to the connection between force and momentum). If one were to write it in terms of the other quantity (the monster) there would be a $\gamma$ factor in this equation so the connection is obscured.

Ultimately all this should not matter very much. There is only one correct prediction about any given physical scenario in special relativity and you can calculate it how you like. Any result whose outcome depends on whether the phrase "3-force" applies to $d{\rm p}/dt$ or $d{\rm p}/d\tau$ is telling us about human language conventions but not about the physical world itself.

A comment on transformation and ease of calculation

By calling $d{\bf p}/d\tau$ a monster I have, I admit, exaggerated a little. It is the spatial part of a 4-vector and as such it is the very thing one is considering in various calculations. Those calculations concern $F^a$ and they are of course perfectly good calculations. But we are also on the lookout for physical insight, and in particular in an experimental (as opposed to pure theory) setting we want insight into what goes on in some inertial frame. It is here that ${\bf f}$ yields more useful insight than $\gamma {\bf f}$, for the reasons given above.

One often hears the view that special relativity ought to be done always in terms of 4-vectors not 3-vectors, and it may be emphasized that equations in terms of 4-vectors are easy to transform between inertial frames. However it often arises that what one really wants to know is what will be observed: where the particle actually goes, what energy will be detected by an instrument, and things like that. For this purpose one has to solve the equations of motion. Now the issue is no longer whether the equations transform neatly between frames; the issue is how to solve them. Some cases come out neatly in terms of proper time, some come out neatly in terms of coordinate time. Neither is innately superior to the other. For example, if one knows at what proper time an accelerating electron arrives at an electrode, one still does not know when it will arrive there! (in terms of the clock in your inertial laboratory). Often we want to know when it arrives in laboratory time. One could multiply examples.