What is the speed/time relationship at constant acceleration?
If you could maintain 1g acceleration for a year how fast would you be going and how far would you get? Whether it is possible for a rocket to accelerate for a year at 1g is another question.
The classical expression is really simple (a and v are vectors):
v = at
s = ½at2
However 1g x 1 year = 1.03c which is greater than the speed of light. Clearly special relativity needs to be taken into account.
I found it a bit confusing, working out what reference frames to use. Special relativity gives transformations for velocities between different inertial (i.e. non-accelerating) reference frames which are travelling at a given relative velocity. However, if a rocket is accelerating at 1g what frame should we use? Acceleration is the rate of change of velocity by time, but what time, since the time measured depends on the reference frame used.
The Lorentz transformation is used to translate acceleration between different inertial frames, as described in Wikipedia
The relevant time to use for an accelerating object is the proper time, the time that the traveller experiences, and the acceleration is called proper acceleration.
The case of constant acceleration is called hyperbolic motion, as the path in an x-t diagram (where x is the direction of the acceleration) has the form of a hyperbola, and the equations describing the motion are given here. Proper acceleration is given a greek letter alpha to align with proper time which is conventionally τ (tau).
The velocity u is given in terms of the inertial frame time T, if u is 0 when T is 0:
I don't know what to do with hypervelotime, type .
Integrating this with respect to T gives the X co-ordinate in the inertial frame.
We can substitute Q = 1+(αT/c)2, because the derivative of Q appears in the numerator of the integrand: so
dQ = (2α2/c2)TdT
I don't know what to do with hyperpositionsub, type .
We can also check when T is small that this approximates to the expressions above for the classical case.
For interstellar voyages the inertial frame is the frame in which Earth, and (with a very small error) the destination star, are at rest.
To find out how long a hyperbolic journey would take, we can invert this expression to give
I don't know what to do with hypertimeposition, type . .
This gives the time observed from Earth for the journey of distance X.
It is a curious co-incidence that if &alpha = 1 earth gravity, 9.81 ms-2, then c/α is only 3% different from a year. Thus αT/c is approximately the time in years and αX/c2 is the distance in light years.
To calculate how long it would take to reach a star, for example Tau Ceti at 11.9 light years, you should use half the distance in the above formula, and then double the resultant time, as (presumably) you want to arrive at Tau Ceti again approximately at rest, so half the journey would be accelerating at 1g, and half the journey would be decelerating at 1g.
By this reckoning getting to Tau Ceti would take 13.8 years. Before you exclaim in confusion that all of the literature talks about thousands of years to get to the nearest star, note that we have calculated the time for constant acceleration all the way there. We do not have the technology nor the energy source to maintain 1g acceleration for anything like 13.8 years.
For comparison with the technology we do have, Voyager 1, currently the furthest man-made thing from Earth, is currently about 20 Tm away, having been travelling since 1977, 46 years, and currently moving at 17 km/sec. Tau Ceti is 113 Pm away, which at that speed would take 218,000 years.
We haven’t quite finished though. As well as the time in the Earth frame of reference we would also like to know the time experienced by the traveller, the proper time τ.
The traveller is not in an inertial frame so we cannot use the Lorentz transformation directly but we can consider a small increment of proper time dτ and co-ordinate time dT, and relate those using the Lorentz transformation for the velocity u it is going at at that point. We have (where γ = γ(u is the Lorentz γ function of the velocity between two inertial frames):
I don't know what to do with propertimedifferential, type .
And substituting the above expression for u(T):
I don't know what to do with propertimediff2, type .
To integrate this, the form of the denominator suggests a substitution using the hyperbolic functions sinh and cosh. Set sinh λ = αT/c. (Hyperbolic functions are included in the Further Mathematics A-level syllabus).
I don't know what to do with propertimeintegral, type .
(noting initial condition τ = T = λ = 0).
Expanding arsinh for those who don’t have it on their calculators:
I don't know what to do with propertimeaslog, type .
There is no doubt that the hyperbolic functions do make manipulating these functions easier. This gives proper time in terms of co-ordinate time T but what about in terms of co-ordinate distance X. If we want to know the proper time for a journey then X is what we would start from.
For this final bit I am going to use hyperbolic functions only, it’s just easier.
Rewriting X(T) from earlier with hyperbolic functions, we can do all that we did before a bit more easily.
I don't know what to do with hyppositiontime, type .
This is quite easy to invert just by unwrapping the functions:
I don't know what to do with hyptimeposition, type .
Now substituting T back in the equation for τ(T):
I don't know what to do with propertimeofposition, type .