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, elapsed, calculate and/or ship:

    “There is a difference between twenty-nine and thirty. When you are twenty-nine it can be the beginning of everything. When you are thirty it can be the end of everything.”
    —Gertrude Stein (1874–1946)

    “The quickness with which all the “stuff” from childhood can reduce adult siblings to kids again underscores the strong and complex connections between brothers and sisters.... It doesn’t seem to matter how much time has elapsed or how far we’ve traveled. Our brothers and sisters bring us face to face with our former selves and remind us how intricately bound up we are in each other’s lives.”
    —Jane Mersky Leder (20th century)

    “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)

    “... what a family is without a steward, a ship without a pilot, a flock without a shepherd, a body without a head, the same, I think, is a kingdom without the health and safety of a good monarch.”
    —Elizabeth I (1533–1603)