Twin Paradox - Difference in Elapsed Times: How To Calculate IT From The Ship

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:

\begin{align} \Delta t^2 & = \left \,\left \\ & > \left^2 = \left^2 = \Delta \tau^2. \end{align}

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:

    “A faith is something you die for, a doctrine is something you kill for. There is all the difference in the world.”
    —Tony Benn (b. 1925)

    “There is all the difference in the world between the criminal’s avoiding the public eye and the civil disobedient’s taking the law into his own hands in open defiance. This distinction between an open violation of the law, performed in public, and a clandestine one is so glaringly obvious that it can be neglected only by prejudice or ill will.”
    —Hannah Arendt (1906–1975)

    “... 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)

    “O Captain! my Captain! our fearful trip is done,
    The ship has weather’d every rack, the prize we sought is won,
    The port is near, the bells I hear, the people all exulting,
    While follow eyes the steady keel, the vessel grim and daring;
    But O heart! heart! heart!
    O the bleeding drops of red,
    Where on the deck my Captain lies,
    Fallen cold and dead.”
    —Walt Whitman (1819–1892)