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Newtonian Force and Work from Special Relativity? Francesco R. Ruggeri Hanwell, N.B. Dec. 26, 2025 It is well known that one may obtain expressions for energy mocc/sqrt(1-vv/cc) and momentum p=mov / sqrt(1-vv/cc) from the special relativistic notion of viewing a particle at rest from a frame moving at constant speed -v. In other words, no notion of force or work is needed to obtain these expressions. We ask: How does the notion of force, work and conservation of energy and momentum then arise from special relativity? We suggest that force and work may be introduced in a purely mathematical manner and a physical interpretation applied later. We first consider why the Lorentz invariant -Et+px (-Et+p dot r) should exist. In particular, we note that t and x are not on the same footing as t→ 0 to infinite, while x → - infinite to infinite. Furthermore, E is a number while p is a vector. We suggest that -Et+px is linked to informational considerations. We have noted this before, but we reiterate it here because we think it is linked to the notion of work. In particular, p dot r is a vector associated with a measure ( r vector) which has two directions, just like the vector. The dot product p dot r, however, is a number, so p to the right with r vector to the right is the same as p vector to the left with r vector to the left. We next consider a rest mass 2mo at rest and compare it with a Lorentz boosted rest mass mo with -v and another with mo and v. In such a case, momentum is the same in the rest frame and in the added two frames moving in opposite directions. This suggests the idea of conservation of momentum, but the same cannot be said about energy. 2mo cannot be boosted to two separate frames, so one does not have energy conservation. If 2mo breaks into mo and mo, one cannot have less rest mass it seems and there is no conservation for 2mo, so one may suggest that for conservation one must have 2mocc + extra internal energy in the rest frame. Thus, the two frames moving with v and -v are created due to this extra internal energy which is a number. From p dot r considerations, however, one may postulate a vector F and use Integral F dot dr = .5 extra internal energy for each particle. In this way the notion of force (which is equal and opposite) and work arise strictly from mathematical consideration. One does not need to know that F is a push or a pull (force). It is simply a math vector linked with dr. Later, one may try to assign it physical meaning. One may then derive Newton’s second law from these considerations as we show. Special Relativity One may obtain equations for momentum and energy by considering a particle with rest mass mo at rest at x=0 at t without any notion of force or work by considering an equivalence between what is seen in the rest frame and what is seen in a frame moving at constant speed -v. In both frames, x=0,t=0 and x’=0, t’=0 represent the initial state. In the moving frame, however x’/t’ = v ((1)) This suggests a Lorentz transformation of:
[ g(v) vg(v) ] ((2)) [vg(v) g(v) ] where g(v) is unknown. Applying this same transformation to p=0, and E=mocc, one has: p’ = g(v) v mocc and E’ = g(v) mocc ((3)) If one wishes to have the two states be equivalent in some sense, one may argue that mathematically they have the same modulus, but with a minus sign metric because both E’ and p’ are larger than their values in the rest frame. Thus: -EE + p dot p= - momo cccc ((4)) This suggests that -Et+ p dot r is also a Lorentz invariant. We wish to consider this invariant from the point of view of information in the next section. Informational Interpretation of -Et+p dot r We have considered -Et+ p dot r from the point of view of information in previous notes, but we wish to reiterate this here because we think it is linked to an idea which may be applied to force dot dr = work, which we discuss later. We note that t and x are not on the same footing, just as E and p are not on the same footing. t goes from 0 to infinite while x (one dimension) goes form -infinite to infinite ((5)) If one says that there is ‘motion’ in time it is clear (one direction), but the statement that there is motion in x is not clear as there are two directions. The same considerations apply to E and p. As a result, if one wishes to combine E,p, t and r in a Lorentz invariant expression which has each variable appear only in linear form, then one may use: -Et + p dot r ((6)) (The minus sign is discussed in the above section.) The use of a vector which may point in either direction dotted with dr leads to a number. In such a way, motion in one direction or the other can lead to the same number. For instance, p with dr positive leads to p dot dr equal to p negative with dr negative as well. There is no preference for an x-axis pointing to the right with motion to the right versus an x-axis pointing to the left with motion to the left. We argue that this mathematical/symmetry notion is applicable to the notion of work.
Energy and Momentum Conservation and the Mathematical Emergence of Work and Force In the first section, we noted how one may consider a particle with rest mass mo at rest at x=0 at t from a frame moving at -v. We now consider two mo’s , one viewed from a frame moving at -v and the other, from a frame moving at v. In such a case, the total momentum is 0 which matches the momentum of a mass of 2mo at rest. The problem is that the energy of the combined 2mo can only be seen from one frame or the other. Energy added for the v and -v frame does not match the energy in the 2mo frame. In order to have energy conservation, one needs to add an energy term to 2mocc, i.e. Energy in the rest frame = 2mocc + extra energy ((7)) This extra energy is a number. One frame represents motion to the right, and the other to the left. This is reminiscent of p dot dr representing a number. In particular, .5 extra energy should apply to each frame and this is the same number for each frame. We argue that one may introduce a mathematical vector called F such that: .5 extra energy = Integral F dot dr ((8)) ((8)) applies for both frames. In such a case, two new math ideas appear, namely F and Integral F dot dr. The question is whether these have any physical relevance. As is known, F may be considered a push or pull and Integral F dot dr, work, but here they arise solely as math constructs, with the physical interpretation being applied later. Newton’s Second Law A particle with rest mass mo at x=0 at t is seen to have energy E = mocc/sqrt(1-vv/cc) and momentum p=mov/sqrt(1-vv/cc) without any notion of work or force being considered. In the above section, however, we introduced the notion of force and work and called work the extra energy seen in the moving frame. Thus: mocc/sqrt(1-vv/cc) - mocc = Integral F dot dr ((9)) Here dr is measured in the rest frame. Given v(r) = dr/dt and considering motion in one direction with F pointing to the right, if one assumes that: F = dp/dt then Integral dp/dt dot dr = Integral F dot dr Integral v dp = Integral F dot dr Set c=1 (one dimension here for simplicity) Integral v(r) dot dp = Integral sqrt(1-vv)/mo p dp but -EE+ pp = -momo (one dimension) So E dE = p dp → Integral dE = Integral F dot dr which is equivalent to ((9)) so
F = dp/dt ((10)) which is Newton’s second law. Conclusion In conclusion, special relativity yields expressions for energy and momentum, mocc/sqrt(1-vv/cc) and mov/sqrt(1-vv/cc) without any notion of force and work appearing.One simply considers a particle with rest mass mo at rest at x=0 at t and the state seen from a frame moving at constant -v. One creates a transformation linking x’,t’ and E’,p’ with E=mocc, p=0, x=-0,t and then argues that physically the states are the same which should be shown by a constant modulus, but with a minus metric. This begs the question: How does the concept of force and work arise in special relativity? We argue that one may consider two mo’s in a rest frame, one being seen by a frame moving with -v and the other by a frame moving with v. Together this is like having two mo’s and a total momentum of 0. One might consider momentum to be conserved, but not energy because 2mocc in a rest frame may only be seen by one frame or the other. To have energy conservation, we argue that one must have 2mocc+ extra energy in the rest frame. In other words, the extra energy is needed to create the extra energy in the two frames, i.e. energy beyond mocc in each frame. Using ideas of p dot dr being the same for motion to the right with an x-axis pointing to the right and -p with an axis pointing to the left, we suggest that one may introduce a math vector F such that .5 extra energy = Integral F dot dr. This holds then for both moving frames. As a result, F and Integral F dot dr are introduced in a purely mathematical manner and one may later assign the physical meaning. We suggest that this how the notion of force and work may be arise in special relativity which is based on viewing a particle from different frames. We also show that Newton’s second law dp/dt = F also holds with dt being the usual dt of the rest frame.
