Relativistic Dead Reckoning
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If we have a starship under constant uniform acceleration, we must use the formulae that we have gained from the Lorentz Transformation to calculate its position in space and time. To aid us in that application we must decide which of the infinite set of inertial frames of reference we will define to play the role of the proper frame, the frame of pseudo-rest. The frame occupied and marked by our sun gives us an obvious Galactic Standard Inertial Frame of Reference, though we must take care to acknowledge the motions of the sun in response to the tug of its orbiting planets and that due to the sun's motion as a component of the Milky Way. Those motions are sufficiently small that we may ignore them here.
To provide an example of an interstellar crossing I will work out the relevant numbers for a voyage from Sol to Procyon, a distance of 11.2 lightyears or 4090.8 lightdays.
In the starship's frame the ship gains dV = AdT, in which A represents the acceleration read off the ship's accelerometers. As seen from the Galactic Standard Inertial Frame of Reference the ship's speed increases from v to (v+AdT)/(1+vAdT/c2), which we obtain by applying the relativistic velocity addition formula. We then get the change in speed from
In that derivation I gave both terms to the right of the equality sign a common denominator, carried out the subtraction, and then reduced the denominator by dropping the dT term, which is vastly smaller than one and, thus, makes effectively no difference in the calculation. Making the substitution β = v/c, we rewrite that equation as
and integrate it to obtain
We now solve that equation for β and obtain
That equation enables us to calculate the speed of a ship relative to whatever inertial frame we have chosen to be the standard, calculating the speed from the time T elapsed on the ship's clocks applied to the acceleration A measured aboard the ship. Of course the principle of Relativity (Einstein's first postulate) tells us that existence has so structured space and time that the ship's crew must see an observer in that standard frame moving at that same speed, albeit in the opposite direction.
In relativistic flight the Lorentz factor tells us more than does velocity, so we calculate from Equation 4
Velocity integrated with respect to time elapsed yields distance crossed. In the standard inertial frame we calculate the ship's progress as x = vdt. Accounting for time dilation, the ship's astrogator calculates
Inverting that equation gives us
We have before us a remarkable path of proof that leads to verification of that equation. Multiply it by the rest mass of the ship and then by the square of lightspeed. We obtain thereby the algebraic statement that the total energy of the ship equals the rest energy plus the work done upon the ship by its acceleration.
On the way to Procyon our ship will gain its maximum Lorentz factor when it has crossed half the distance to its destination, 2045.4 lightdays out from Sol. We have for that Lorentz factor
Typically a starship will accelerate halfway to its destination and then decelerate the other halfway. Because there are no signposts in space, the ship must calculate the time of its thrust reversal. Knowing how far the ship must go, the astrogator calculates the Lorentz factor that the ship will gain by the halfway point and then calculates the time from that datum. The equation to be used comes from multiplying Equation 5 by exp(AT/c), solving the resulting quadratic equation, and then taking the logarithm of the solution. That procedure gives us
On our voyage to Procyon that time works out to
An observer in the standard frame would also like to calculate the time that will elapse on their clocks before the ship reaches the turnover point. Accounting for time dilation and substituting for the Lorentz factor from Equation 5, they calculate
When we plug in the appropriate numbers and carry out the calculation we find that half the journey takes 2372.889 days (6.497 years) in the Galactic Standard Frame or 4745.778 days (12.993 years) for the entire journey. Compare those figures to the 1840.712 days (5.04 years) that elapse on the ship's clocks during the voyage.
Those are the equations needed for basic relativistic dead reckoning.
= 9.81 meters per second per second
= 847.584 kilometers per second per day
= 0.0028272359 cee per day (A/c)
= 0.0028272359 added to Lorentz factor per lightday crossed
= 353.702469 days per cee (c/A)
With respect to the standard inertial frame, the ship's error in distance crossed under uniform acceleration is
That result came from Equation 6 by pseudo-differentiation, which I have represented by a lower-case delta to indicate that this is not a standard differentiation, though it uses the same process. We calculate the error in the Lorentz factor by pseudo-differentiating Equation 5 with respect to acceleration to obtain
Combining Equations 10 and 11, we can then calculate the allowable error in acceleration given an allowable error in position,
That equation only applies to trajectories traced by a body under uniform acceleration, but we can apply it to a typical starship trajectory by assuming that the error in distance is cumulative over the two halves of the voyage.
We can also invert the equation and calculate the error in position from a given error in acceleration (the amount by which the ship's actual acceleration differs from the acceleration used by the astrogator). If the error in acceleration equals one one-millionth of a gee, then the error in our voyage to Procyon amounts to 319 million kilometers, roughly two AU.
Thus we have the equations that we need to carry out the deduced reckoning of the trajectory of a starship, presumably a Bussard ramjet, flying through interstellar space under uniform acceleration in its own frame.
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