Hooke's Law: F = -kx for a spring with spring constant k and displacement x.
x is a spatial displacement so this is not relativistic. We want to describe a spring in a Minkowski spacetime.

If we imagine a spring that is several light years long then we can wiggle one end of the spring and it will be quite a while before the other end can possibly observe an effect. There is relevant information in the spring, not just in the position of the end points. There are compression waves and transverse waves. With appropriate choice of coordinate system we can address both waves at the same time.

To describe a compression wave on a spring we can take a spring that is at rest with uniform tension and then mark off constant intervals (e.g. centimeters). Likewise, for a flat elastic medium at rest with uniform tension we can mark off a coordinate grid. Then compression waves are visible as compression of the coordinates. What we are doing is designating a coordinate chart on the elastic medium. Naturally, the coordinate chart should also include the time axis. We will use y_B to for the coordinates on the elastic medium. The index B is a capital letter, to distinguish these coordinates. Write the coordinates of the Minkowski spacetime as x_a. Note that we will have tensors that use both the coordinate system y_B and the coordinate system x_a. Capital indices indicate the coordinate chart on the spring, y_B. Lower case indices indicate the coordinate system in the Minkowski space, x_a.

We will use the function F_a(y_B) to describe the position of the elastic medium in spacetime. For a position and time on the spring at rest, designated y_B, the function F_a(y_B) gives the position and time of that point in the Minkowski space. We are embedding a manifold, the elastic medium, into a flat space of higher dimension.

We suggest the Lagrangian L = (F_{a,B} F^{a,B})^{1/2} as a starting point.

When we integrate the Lagrangian, we are integrating over the elastic manifold, using its coordinate system.

∫ (F_{a,B} F^{a,B})^{1/2} d^n y_B

This is important to maintain the appropriate density of the spring.

If the spring has 0 spacelike dimensions (it is a point) and one timelike dimension, then we are mapping a line into a Minkowski space and the Lagrangian should be familiar. In this case the Lagrangian is

L = (F_{a,B} F^{a,B})^{1/2} with B= 0 and this is
L = (v_a v^a)^{1/2} for v_a = F_{a,0}
and v_a is the 4-velocity as parametrized by the B coordinate. So we have an extra degree of freedom. This is the same problem as stretching a spring around corners. The optimal solution is for the spring to have uniform tension and to follow the path of shortest length. In the case of (F_{a,B} F^{a,B})^{1/2} the optimal solution is for the parametrization to be uniform with respect to arc length and for the path to optimize the proper time. The Lagrangian assures the parametrization is uniform. This makes F_{a,0}F^{a,0}=1 for the particle, the parametrization matches the proper time. To convert to Minkowski space coordinates we set v_0 = 1 (equivalently, F_{0,B}F^{0,B}=1) and then

L = 1/γ which is the Lagrangian for a pointlike particle.

Looking at L = (F_{a,B} F^{a,B})^{1/2} we see that this does not have all the information we need. There is no way to designate the density of the elastic medium or its spring constant. To fix that we change to

L = (F_{a,B} F^{a,C} S^B_C)^{1/2}

Integrating over the elastic manifold we have the Lagrangian
∫ (F_{a,B} F^{a,C} S^B_C)^{1/2} d^n y_B

for some tensor S^B_C. If we assume an isotropic elastic medium with uniform elasticity and density d then we can assign S^0_0 = d^2 and S^N_N = -k for N non-zero.

For a one-dimensional spring of unit length that is stretched by a factor of r we get

L = (F_{a,0}F^{a,0} d^2 + F_{a,1} F^{a,1}(- k))^{1/2}
the term F_{a,0}F^{a,0} d^2 = d^2 but in Minkowski space coordinates it is (d/γ)^2. This term is much larger so we have approximately

L = ((d/γ)^2)^(1/2) = d/γ
Integrating this over the length gives m/γ so that's good. (We are integrating over y_B so the length is still 1 even though the spring is stretched in Minkowski space). We can expand (F_{a,0} F^{a,0} d^2 + F_{a,1} F^{a,1}(-k))^{1/2} as a Taylor series. We assumed a uniform expansion by a factor of r, so (-k)F_{a,1}F^{a,1} = k r^2. The Taylor series adds a factor of (1/2). Therefore, integrating over the spring's unit length gives a potential energy contribution of (1/2)k r^2. So the weak and slow case matches Hooke's law.


To put masses on the end of the spring we can set the values of S^B_C(y_B) so that there is mass on the ends. This introduces the possibility that S^B_C changes as a function of location on the spring. We can include that in the derivation of the equation of motion.

Deriving the equation of motion for L = (F^{a,B} F_{a,C} S_B^C)^{1/2}:

taking the derivatives w.r.t. F_{a,C}

(1/2)(F^{a,B}S_B^C + F^{a,B}S^C_B}) (F^{a,B} F_{a,D} S_B^D)^{-1/2}

(and changing a pair of C indices to D at the end so that things will look cleaner later)

taking the derivative with respect to y_C (and removing the 1/2 factor)

[F^{a,B}_{,C}S_B^C + F^{a,B}S_B^C_{,C} +
F^{a,B}_{,C}S^C_B + F^{a,B}S^C_{B,C}] (F^{a,B} F_{a,D} S_B^D)^{-1/2}
-
(1/2)(F^{a,B}S_B^C + F^{a,B}S^C_B)(F^{a,B} F_{a,D} S_B^D)^{-3/2}
[F^{a,B}_{,C}F_{a,D}S_B^D + F^{a,B}F_{a,D,C}S_B^D +
F^{a,B}F_{a,D}S_B^D_{,C}] = 0


which, frankly, is a little complicated. I prefer the Lagrangian.

There are several interesting features here. One is that you parametrize the path in spacetime, even for a particle. That same effect relates to the relative time dilation experienced on different parts of the spring. The time coordinate in F^{a,B} is a=0 and it follows its own approximation of the wave equation on the spring. Another interesting feature is the two different coordinate systems that are used together. The coordinate systems can be considered a description of the elastic medium geometry in spacetime but it is also convenient to think of the Minkowski space coordinates as a field on the manifold of the elastic medium.