The Relativity of Force

We accept force as one of the fundamental concepts of physics. Indeed, it goes back to the beginning of modern mathematical physics in the Seventeenth Century. Defining and describing it through his three laws of motion, Isaac Newton used it to transform the static geometry of the Ancient Greeks into the dynamic geometry that we call classical mechanics. Over the subsequent centuries physicists parlayed force into additional concepts, such as action and energy. Now physicists tend to use those concepts instead of force, but it still comes up often enough that we need to devise a relativistic treatment of the concept.

In his first law of motion Isaac Newton effectively defined force as that which causes a body to deviate from uniform motion in a straight line. In his second law of motion he refined that definition by equating the strength of the force applied to a body to the rate at which that body’s linear momentum changes. And in his third law he stated the reciprocity relation between force-exerting bodies. With those laws he could then interpret the motions of bodies and deduce the mathematical description of the forces exerted upon them, as he did in devising his law of gravity. Mathematically the second of those laws comes to us as

(Eq’n 1)

in which the product of the forced body’s mass and velocity, (m**v**),
represents the body’s quantity of motion (linear momentum). We want to convert
that equation into a detailed relativistic form.

In order to obtain the correct description of force in relativistic situations, we must acknowledge that the concept of force shows us two aspects. We have the applied force, the force that comes from one body onto another, represented on the left side of Equation 1. And we have the inertial reaction, the description of how a body’s linear momentum changes in response to an applied force, represented on the right side of Equation 1. We must analyze both forces in the light of Relativity before we can claim to have a correct description of forced systems moving at relativistic speeds.

The Inertial Reaction

We define the measure of the force applied to a body as the rate at which the linear momentum of the body changes with the elapse of time. To work out a relativistic version of that definition we need to work out a relativistic description of linear momentum.

In the essay "Four-Vector Kinematics" we found that differentiating the distance-duration 4-vector measured in a given inertial frame with respect to the elapsed time measured in that frame does not produce a proper 4-vector. The 4-component measured-velocity vector fails to conform to Minkowski’s theorem, the 4-dimensional analogue of the Pythagorean theorem: its square does not remain invariant under the Lorentz Transformation. However, we can convert the measured velocity into a proper 4-vector by multiplying it by the Lorentz factor between two inertial frames between which we want to transform the vector.

If we multiply the 4-component velocity of a particle by the particle’s relativistic mass, the product of the particle’s rest mass and the Lorentz factor between the rest frame and the measured-velocity frame, then we automatically get a proper 4-vector. Thus we infer that the 4-dimensional linear momentum of the particle,

(Eq’n 2)

conforms to Minkowski’s theorem and, thus, has its own version of the Lorentz
Transformation. By convention we call {P} the formal momentum of the particle
and call **p** the physical momentum.

As an aid in working out that transformation, imagine the existence of two inertial frames of reference, each marked by a 4-dimensional Cartesian coordinate grid, which grid we may lay out, according to Einstein, with a simple array of rulers and clocks. Imagine further that the two frames move relative to each other at a speed V oriented parallel to the frames’ common x-axis and that the event (x,y,z,ict)=(0,0,0,0) in one frame coincides with the event (x’,y’,z’,ict’)=(0,0,0,0) in the other frame. Subsequent measurements of distance and duration to some other event made by observers in one frame can be translated into copies of what observers in the other frame would measure by way of the Lorentz Transformation.

To devise the Lorentz Transformation of the linear momentum 4-vector we will exploit the theorem of which Minkowski’s theorem shows us one part. But first we want to simplify our task by eliminating two of the spatial dimensions from consideration. We do that by exploiting the conservation law pertaining to linear momentum.

If our two teams of observers measure the linear momentum of a single body, they know immediately that the y- and z-components have the same value in both frames,

(Eq’ns 3)

We know that those equations stand true to Reality because we know that if anyone were to accelerate from one frame to the other in the x-direction, that acceleration will not produce a force in the y- and z-directions; thus, in accordance with the theorem of conservation of linear momentum, the components of the body’s linear momentum in those directions do not change, so those components have the same value for both of our teams of observers as shown in Equations 3.

To work out the Lorentz Transformation of linear momentum in the x- and t-directions, our observers exploit the fact that taking two 4-vectors based on measurements made in the same inertial frame and multiplying them together in the manner of an inner product yields a Lorentz scalar, which remains invariant under a Lorentz Transformation. They pick two events separated from each other only in the x-direction and in time, events that the body under study touches. If the body consists of a single atom, then the events might consist of the atom absorbing a photon and the atom re-emitting the photon. Both teams of observers can then calculate the action that the body enacts between those events,

(Eq’n 4)

They already know the Lorentz Transformation for distance and duration,

(Eq’ns 5)

so they need only make the appropriate substitution into Equation 4 to get

(Eq’n 6)

By equating terms with common spatio-temporal factors across the equality sign we obtain the equations of the Lorentz Transformation for linear momentum,

(Eq’ns 7)

Thus we get the Lorentz Transformation for linear momentum.

Now we want to do the same for force. Suppose that one
team of observers measures the force **f’**=(f_{x}’,f_{y}’,f_{z}’)
acting on a body that ponders a mass m. At one instant the body comes
temporarily to rest, so at that instant the observers’ frame becomes the proper
frame for this analysis. The other team, moving relative to the first, can’t
measure the force directly, so they ask whether those measurements represent the
spatial components of a 4-vector. If they don’t, then the observers can’t use
the Lorentz Transformation. How, then, can the second team transform the first
team’s measurements into numbers that they can use?

We begin with the assertion that differentiating a 4-vector with respect to a Lorentz invariant yields another 4-vector. In this case, in accordance with the definition given in Equation 1, we differentiate the 4-momentum with respect to the proper time. But we can’t differentiate the components of the momentum with respect to proper time; we must convert the proper time to the measured time, so we get

(Eq’n 8)

In that equation we use, for convenience, the Greek letter gamma to represent the Lorentz factor between the proper frame and the frame in which observers make the measurements reflected in the terms inside the parentheses. That equation gives us the relation between formal force and physical force.

In order for {F} to represent a true 4-vector its square
must give us a Lorentz scalar, a number that remains invariant under the Lorentz
Transformation. That means that the square of {F’}=γ(f_{x}’,f_{y}’,f_{z}’,0),
in which γ=1,
must equal the square of Equation 8;

(Eq’n 9)

If we take into account the fact that dE/dt=f_{x}V, we see that we
must necessitate that

(Eq’ns 10)

To prove and verify the validity of those equations we need only apply the Lorentz Transformation to {F’} and compare the result with Equation 8. Carrying out that procedure gives us

(Eq’n 11)

As usual, again for convenience, we have β=V/c. In making the comparison we have

(Eq’n 12)

Equating comparable terms gives us Equations 10 and dE/dt=f_{x}V.

Thus we obtain a description of the relativity of force.

The Applied Force

Mathematically this aspect of force gives us the more difficult challenge. I don’t want to take something like Coulomb’s law and naively apply the Lorentz Transformation to the distance in it. Instead, I want to devise some analysis that will reveal clearly how relative motion affects the pushes and pulls that bodies exert upon each other. Devising that analysis will provide the subject matter of further essays, a separate essay for each of the forces we encounter.

Aberration of Force

We often assume that, because we represent forces as being mediated by photons or other radiation particles, that they follow the same law of aberration that applies to light. But if that were true to Reality, then forces acting between moving bodies would violate the conservation laws. We need to take another look at forces to discern what really happens to them in moving systems. Again, that need points to further essays.

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