The problem is this: You board a spaceship that accelerates at
1 "gee" continuously until it has traveled half of the 2.2 million
lightyear distance to the Andromeda Galaxy*. It then decelerates
(also at 1 "gee") over the remaining 1.1 million lightyears of distance
to halt in a star system with an earth-like planet orbiting a
sol-like star in the Andromeda galaxy itself! How much older are
you at the end of the trip? You may be surprised.
Notes below outline (but do not complete)
a novel solution to this problem that makes use of
Galilean chase-plane time T, and Galilean-kinematic velocity V = dx/dT.
These obey Galileo's equations for constant (proper)
acceleration at any speed, even the classical
expression for kinetic energy K=½mV2.
A solution in terms of the more widely used (and useful)
coordinate-velocity v = dx/dt may instead be found
In both cases, we also find it useful to define from Minkowski's
spacetime Pythagoras' theorem, i.e. the
metric equation: (c dτ)2 = (c dt)2 - (dx)2,
the proper or traveler time τ and the proper velocity w = dx/dτ.
The "one map two-three clock" equations of anyspeed acceleration are summarized
Intro to the variables used here:
this PDF or
If Andromeda is too far away, consider
race to Alpha-Centauri!
Background: Because of the high speeds involved, only clocks in
Galileo's equation-saving "chase plane" follow the time parameter
T used in Galileo's classical acceleration equations.
Instead, the rocket or "proper-time" dτ elapsed for you on
each of the two constant acceleration legs of the trip is
dτ=(ArcSinh[wf/c] - ArcSinh[wo/c])*c/α,
where the speed of light c is 3x10^8[m/s2] or 1[ly/yr],
proper acceleration α is 9.8[m/s2] or
wf & wo denote final & initial
proper velocities (w = dx/dτ) in units of "distance
traveled per unit of
traveler time". Relativists will recognize proper velocity
w as c Sqrt[γ2-1], where
γ = E/mc2. Proper velocities here can be
figured by the conversion
w = V Sqrt[1 + ¼( V/c )2],
from "Galilean velocities" Vf and Vo which
obey the classical equations for constant acceleration. The standard
Vf2 - Vo2 = 2 α dx
(where dx is distance traveled) in particular should do the job.
Given initial/final proper velocities, the maptime dt elapsed
is simply (wf-wo)/α,
and coordinate velocity v=dx/dt
is w/γ. Can you show that finite proper velocities w
require that v is always less than c?
Note: The "Galilean velocity" V is the velocity familiar
from introductory physics books extended to all velocities as
Galileo might have presumed through the kinetic energy equation
K = (1/2)mV2. It thus becomes the derivative of x with
respect to a "Galilean chase-plane time" T which can be used to
track events during 1D acceleration, but which at high speed follows
neither traveler nor earth based clocks. At high speeds
Galilean velocity V is not the Lorentz/Minkowski coordinate
velocity v = dx/dt.
* The Andromeda Galaxy is one of the most distant objects visible to
the naked eye. Total distance traveled is
2.2x106 [ly] x 9.46x1015 [m/ly] = 2.08x1022 [meters].
Followup: Do you like doing relativistic kinematics in
this way? More examples now under construction, along with some answers, will be put here.
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