# Four-Vectors in One-D Acceleration

Supplementary notes for a class on modern physics
Relativity is more elegantly expressed in terms of 4-vectors, than in terms of the scalar and 3-vector variables familiar from introductory physics. Hence everything discussed on these acceleration pages can be put into 4-vector form as well. We do this here for those interested. The traveler kinematic is favored, where by traveler we mean someone who judges distance from the context of a frame with respect to which she is moving.

The "four-vectors" for our "1D acceleration problems" of course have zero y and z components. We will therefore only discuss their coordinate time t and their position x components. We then might write the four components of a space-time displacement four-vector as X={ct,x,0,0}. The magnitude or length of this 4-vector, which we might denote as cτ where τ (tau) is a frame-invariant proper time, refers to the square-root of the scalar product: X•X = (ct)2 - x2 - y2 - z2 = (c τ)2. The negative signs in the scalar product sum extend Pythagoras theorem to include time, with the result that proper time intervals between events can be real (sum>0) or imaginary (sum<0). In this latter case, some call the proper time interval "spacelike" instead of "timelike".

The velocity four-vector associated with this displacement 4-vector is U = dX/dτ = {cγ,w,0,0} = {c cosh[η], c sinh[η], 0, 0} where γ is energy factor gamma or dt/dτ, w is proper velocity dx/dτ, and η is the hyperbolic velocity angle eta or rapidity. The magnitude of this four vector is lightspeed c, which if multiplied by rest mass m, by c, and then squared, verifies (or follows from) the relativistic mass-energy equation (mc)2U•U = (mc2)2 = (γmc2)2 - (mwc)2 = E2 - (pc)2 where p=mw is relativistic momentum and E=γmc2 is relativistic mass-energy. If p=0, this is the familiar E=mc2.

The acceleration 4-vector associated with this displacement 4-vector is A = dU/dτ = {c dγ/dτ, dw/dτ, 0, 0} = {c sinh[η] dη/dτ, c cosh[η] dη/dτ, 0, 0}. The magnitude-squared of this 4-vector is thus a negative number, which we can define as -α2 where α is the instantaneous proper acceleration. In other words, A•A = -α2 = -(c dη/dτ)2, so that dη/dτ = α/c. The proper time derivative of eta, times c, is thus the instantaneous proper acceleration.

If we multiply the acceleration 4-vector by rest mass, we get the relativistic 4-vector version of Newton's law, F = mA = m dU/dτ = {(1/c)dE/dτ, dp/dτ, 0, 0}. Note that if force is zero, the spatial part of this 4-vector expresses conservation of momentum, as is true for the 3-space version of Newton's law. An added bonus with this 4-vector form is that the time-component of the vector (listed first) expresses conservation of energy.

None of the foregoing assumes that this proper acceleration is constant. Even so, each of these 4-vectors can be written in terms of the velocities associated with any of the three kinematics: inertial, Galilean, or traveler, using conversions derived elsewhere. For example, Galilean-kinematic velocity of our traveler is defined as V ≡ dx/dT where T is time on the clocks of chaseplane whose motion allows one to describe the traveler's proper acceleration with Galileo's equations e.g. α = dV/dT. In these terms the vector 4-velocity obeys U = {c (1+½(V/c)2), V Sqrt[1+¼(V/c)2], 0, 0}. This equation thus gives Galileo's acceleration equations predictive power in the relativistic regime.

If proper acceleration α is constant, we can further integrate these equations to predict A, U, and X as a function of time. To make life simple, as in the universal acceleration plots discussed elsewhere on these pages, we write the result for the case when the zero of time corresponds to the rest (zero-velocity) point along the constant acceleration trajectory. We get η = ατ/c, A = {α sinh[ατ/c], α cosh[ατ/c], 0, 0}, U = {c cosh[ατ/c], c sinh[ατ/c], 0, 0}, and X = {sinh[ατ/c] c2/α, (cosh[ατ/c] - 1) c2/α, 0, 0}, or... Again, these can be re-expressed in terms of time in any of the three kinematics. Note in particular that the components of A are not constant, but are undergoing uniform "hyperbolic rotation" in the velocity angle η, as the accelerated object speeds up.