Difference in Elapsed Times: How To Calculate It From The Ship
In the standard proper time formula
Δτ represents the time of the non-inertial (travelling) observer K' as a function of the elapsed time Δt of the inertial (stay-at-home) observer K for whom observer K' has velocity v(t) at time t.
To calculate the elapsed time Δt of the inertial observer K as a function of the elapsed time Δτ of the non-inertial observer K', where only quantities measured by K' are accessible, the following formula can be used:
where aτ is the proper acceleration of the non-inertial observer K' as measured by himself (for instance with an accelerometer) during the whole round-trip. The Cauchy–Schwarz inequality can be used to show that the inequality Δt > Δτ follows from the previous expression:
Using the Dirac delta function to model the infinite acceleration phase in the standard case of the traveller having constant speed v during the outbound and the inbound trip, the formula produces the known result:
In the case where the accelerated observer K' departs from K with zero initial velocity, the general equation reduces to the simpler form:
which, in the smooth version of the twin paradox where the traveller has constant proper acceleration phases, successively given by a, −a, −a, a, results in
where the convention c = 1 is used, in accordance with the above expression with acceleration phases Ta = Δt/4 and coasting phases Tc = 0.
Read more about this topic: Twin Paradox
Famous quotes containing the words difference in, difference, elapsed, calculate and/or ship:
“The sadistic person is as dependent on the submissive person as the latter is on the former; neither can live without the other. The difference is only that the sadistic person commands, exploits, hurts, humiliates, and that the masochistic person is commanded, exploited, hurt, humiliated. This is a considerable difference in a realistic sense; in a deeper emotional sense, the difference is not so great as that which they both have in common: fusion without integrity.”
—Erich Fromm (1900–1980)
“The difference between objective and subjective extension is one of relation to a context solely.”
—William James (1842–1910)
“... most Southerners of my parents’ era were raised to feel that it wasn’t respectable to be rich. We felt that all patriotic Southerners had lost everything in defense of the South, and sufficient time hadn’t elapsed for respectable rebuilding of financial security in a war- impoverished region.”
—Sarah Patton Boyle, U.S. civil rights activist and author. The Desegregated Heart, part 1, ch. 1 (1962)
“However others calculate the cost,
To us the final aggregate is one,
One with a name, one transferred to the blest;
And though another stoops and takes the gun,
We cannot add the second to the first.”
—Karl Shapiro (b. 1913)
“To place absolute trust on another human being is in itself a disaster, both ways, since each human being is a ship that must sail its own course, even if it go in company with another ship.”
—D.H. (David Herbert)