Inertial Navigation and Relativity

I conjure you, by that which you profess,

Howe'er you come to know it, answer me:

though waves confound and swallow navigation up;

though germens tumble all together,

even till destruction sicken; answer me...

                                        Shakespeare

Spatial position is the integral of velocity, which is the integral of acceleration, so in principle from our sensed acceleration we can infer our velocity and position. This is the basis for inertial navigation systems. For fairly low speeds and accelerations (and brief intervals), the equations of Newtonian mechanics give adequate results, but for more precision – especially when very high speeds are involved – it is necessary to use the relativistic equations. In that case we need to determine the time coordinate as well as the space coordinates, since the proper time along the path differs from the time coordinate. For full motion in three space dimensions it’s most efficient to simply numerically integrate the basic equations, but for the special case of motion in a single spatial direction we can easily derive the exact analytical expressions. Derivations have appeared in the literature, such as Minguzzi’s 2004.

For any system s of inertial coordinates x,t, consider a world line with velocity v=dx/dt at a given event, and let S be another system of inertial coordinates X,T in terms of which the velocity V=dX/dT of the world line at that event is zero. Also, let a=dv/dt be the acceleration in terms of s, and let A=dV/dT be the acceleration in terms of S. We saw in “The Inertia of Energy” that

Notice that T, the time coordinate of the momentarily co-moving inertial coordinates at the given event, advances at the same rate as the proper time τ along the world line at that event, so we have dT/dτ = 1 at each event along the world line. We also have dτ/dt = √(1–v²), so for each event on the world line we have A = dV/dT = dV/dτ, and the above relation with A and v as functions of the proper time τ along the world line can be written as

Solving for v, we get the following expression for v as a function of proper time based purely on the profile of proper acceleration A(τ):

We also have the relations

Therefore, making use of the previous expression for v(τ), we have the definite integrals

These equations enable us to keep track of our coordinates for any specified system of inertial coordinates, given our initial speed v(0) in terms of those coordinates and our proper acceleration A(τ) as a function of our proper time. The proper time and proper acceleration can be measured by carrying an ideal clock and accelerometer.

Now consider the invariant time-like interval between x(0),t(0) and x(τ),t(τ) along a given time-like world line, given by

As Minguzzi (2004) noted, since this has the same value (for a given profile of proper acceleration as a function of proper time, which is invariant) in terms of any system of inertial coordinates, we can without loss of generality compute the magnitude of this interval for the inertial coordinate system in terms of which v(0) = 0. Hence the invariant magnitude of the interval can be expressed purely in terms of the invariant proper acceleration profile along an arbitrary path between the endpoints of a time-like interval as

Of course, the dependence on v(0), although not explicit, has not “disappeared”, it is encoded in the acceleration profile for the arbitrary path connecting two endpoints of an interval. For example, if we consider a family of inertial world lines emerging from the origin of inertial coordinates at various speeds, abruptly reversing their speeds (during, say, a one-second time interval) at some time such that they all arrive back at the origin together, the accelerations during those one-second turn-arounds are roughly “twice the initial speed” per second.

Factoring the above expression into a product of the sum and difference of the integrals, and making use of the trigonometric identities sinh(z) = (e^z – e^–z)/2 and cosh(z) = (e^z + e^–z)/2, the squared invariant interval can be written as

The two factors on the right side are not generally equal, since v(0) may not equal zero, which makes the situation directionally asymmetrical. However, if v(0) = 0, meaning the world line is initially tangent to the interval, then the situation is directionally symmetrical, meaning the same profile of acceleration with the opposite sign would yield the same integrals. Indeed for a round trip in terms of the x,t system we have x(τ)–x(0) = 0 and hence the integral of the hyperbolic sine vanishes, and the two factors on the right side of the above expression are equal (as required by symmetry under reversing the sign of A). In this special case, i.e., a round trip with zero initial velocity in terms of the rest frame of the interval, we have

To illustrate, consider a case with constant proper acceleration, resulting in hyperbolic motion, as depicted in the figure below (for a stationary interval).

We consider the portion of the hyperbolic world line that begins and ends at x = 0. By equation (2) we have

The elapsed time coordinate Δt for that hyperbolic world line is

Incidentally, Minguzzi’s derivation of (1) proceeded by letting v=dx/dt, u=dx/dτ, and noting that dt/dτ = 1/√(1–v²). Thus u = (dx/dt)(dt/dτ) = v/√(1–v²). Now, we know the quantities dt/dτ and dx/dτ are the components of the four-velocity, and the derivatives of these with respect to τ are the components of the four-acceleration, whose magnitude A is therefore

in agreement with (1). However, this pre-supposes knowledge of four-vectors and their magnitudes given by the Minkowski inner product which, though correct, is not particularly intuitive. A satisfactory derivation must explicitly establish the proper acceleration as the acceleration in terms of the momentarily co-moving system of inertial coordinates.

Return to MathPages Main Menu