On Required Classical Force Law Adjustments
Newton’s 2nd law states that
In words, the net external force acting on a particle equals the particle’s time rate of change of momentum. F in turn is specified by one force law or another. For example, if the particle is on the end of an anchored spring that is oriented parallel to the x axis, then Hooke’s law specifies that
Here Dx is the amount the spring is stretched or compressed from its relaxed state, and k is a constant associated with the particular spring. Substituting in Eq. 1 produces
In general the Newtonian paradigm is: (1) Use the appropriate force law to determine what F is in a given case; (2) Apply Newton’s 2nd law to determine what d(mv)/dt will be at every point. The particle’s motion can then be predicted either analytically or by using a computer.
A useful and key feature of the Newtonian paradigm is that it works in all inertial frames. For example, a particle might go in a circle on the end of a spring whose other end is anchored in inertial frame K. Application of the laws in frame K’ indicates that the particle will move along a cycloid trajectory. And this trajectory is identical to the path specified by the Galilean transformation. In general, since the Galilean transformation produces a unique path in K’, it places a well-defined requirement on any force law: A force law must produce the correct motion in any inertial frame when it is equated to d(mv)/dt.
With the advent of Maxwell’s equations it was evident that the propagation speed of electromagnetic waves must be independent of the motion of their source. The equations suggest that such waves propagate in all directions with the one speed c. This of course begs the question "one speed c relative to which inertial frame of reference?" For a time it was assumed that the waves are undulating stresses in a medium dubbed "the ether." And the ether was postulated to be at rest in the unique frame where the propagation speed is single-valued.
Such being the case, the Galilean transformation indicates that the propagation speed will vary in different directions, in other frames. Believing that the earth must be moving in the ether frame, Michelson and Morley devised an experiment to ascertain what the motion might be. Their results stunned the physics community: it appeared that the earth is at rest in the ether frame, notwithstanding its annual orbit around the sun.
Fitzgerald suggested that the null results obtained by Michelson and Morley can be explained by a contraction of rigid objects that move through the ether. Lorentz added to this the hypothesis that physical processes slow down when a system moves through the ether. Einstein then pointed out that, when distributed clocks in a given frame are synchronized using the idea that light travels at the one speed c in that frame (as experiments indicate is the case), then different inertial observers will disagree on the times of events.
Length contraction, time dilation and this relativity of simultaneity gave rise to a new transformation, the Lorentz transformation. In the case of the particle going in a circle in frame K, application of the Lorentz transformation results in a quasi-cycloid in K’. That is, in K’ the particle passes "above" and "below" the (now moving) anchor point at the usual distance, but it passes "in front of" and "in back of" the anchor point at a contracted distance. Furthermore, the time for one quasi-cycloid cycle in K’ is greater than the time for one circular orbit in K.
Such peculiarities of the Lorentz transformation suggested that the classical force laws (notably Hooke’s law and Newton’s law of gravity) required modification. The notable exception was the force law of Maxwell/Lorentz. The electromagnetic force, as prescribed by that law, produces the same motion as that predicted by a Lorentz transformation provided only that a particle’s mass is actually a function of its speed.
Combining the dependence of mass on speed with the Lorentz transformation, one can also derive how the components of d(mv)/dt transform. The result is more complicated than the Galilean result that F’=F. But the new "relativistic" transformation has the advantage that it is quantitatively equivalent to the transformation of Maxwell/Lorentz. Indeed it might be stated as a requirement that every force law must transform in a manner consistent with the relativistic transformation of d(mv)/dt.
For example, in frame K (say the frame in which one end of a spring is anchored), Hooke’s law is
The relativistic force transformation indicates that
but length contraction indicates that
Evidently k is not actually a constant. When the anchor point moves,
In words, a spring that moves parallel to its length is stiffer than when it is at rest. By the same token, if the spring moves perpendicular to its length then
Such springs are not as stiff as they are when they are at rest. If these ideas are incorporated into the thought experiment of a particle on an anchored spring , then application of F’=d(m’v’)/dt’ produces the correct quasi-cycloid in frame K’.
An obvious question at this point is, "What about Newton’s force of gravity?" Newton stated that this force is always attractive and such that
This force produces a true cycloid in K’, in accordance with the Galilean transformation. But (like Hooke), it must be modified to produce the quasi-cycloid specified by the Lorentz transformation. Since Coulomb’s law is inverse square, and since Maxwell/Lorentz is consistent with the relativistic force transformation, an obvious modification to Newton’s gravity law is to suppose that there is a magnetic analog in mass-mass interactions.
Such a magnetic analog may have subtle effects (aside from producing the correct trajectory in K’). For example, it is clear that the speed dependence of inertial mass will cause a planet’s elliptic orbit to precess. But this precession is somewhat less than what is observed. However, the sun rotates and accordingly should exert a "B-analog" force, in addition to the usual gravitation force, on a planet moving in its equatorial plane. And since the planets orbit the sun with the same sense as it rotates, the precession of the combined gravitational and B-analog forces should be more than the speed dependence of m alone suggests.
Newton himself suggested that light is particulate, and that such particles would be deflected when passing through a gravitational field. Here again they might also be deflected by a powerful B-analog field. Indeed in the "gravito-magnetic" field of a super-dense, rotating mass, they might be able to escape only along the axis of rotation … a characteristic of "black holes."
Suffice it to conclude here that every force law must be such as to produce results consistent with the Lorentz (i.e. relativistic) force transformation. Whereas it turns out that Maxwell/Lorentz requires no modification, the classical laws (notably Hooke and Newtonian gravity) must be modified. With regard to gravity, a simple problem in statics might provide a suitable conclusion. Fig. 1 depicts two massive, charged spheres. Matter and charge are distributed evenly in each sphere. In rest frame K the spheres repel (Coulomb) electrostatically and they attract (Newton) gravitationally. The charge/mass ratios are just what is required to keep the spheres at rest. That is, according to Newton the net force on each sphere is zero.
According to the relativistic force transformations, the net force must be zero in every other inertial frame. Yet the Lorentz force clearly varies from frame to frame. The mass-mass force must vary identically in order for the net force, F’, to be zero in every inertial frame.
Charge/Mass Spheres in Equilibrium