The 4-acceleration of a particle or other object is defined by a(vector) du(vector)/d, where u(vector) is

The 4-acceleration of a particle or other object is defined by a(vector) ≡ du(vector)/dτ, where u(vector) is its 4-velocity and τ is proper time along its world line. Show that, if an observer carries an accelerometer, the magnitude |a| of the 3-dimensional acceleration a measured by the accelerometer will always be equal to the magnitude of the observer’s 4-acceleration,

(b) In the twins paradox of Fig. 2.8a, suppose that Florence begins at rest beside Methuselah, then accelerates in Methuselah’s x-direction with an acceleration a equal to one Earth gravity, g, for a time T_Florence/4 as measured by her, then accelerates in the −x-direction at g for a time T_Florence/2, thereby reversing her motion; then she accelerates in the +x-direction at g for a time T_Florence/4, thereby returning to rest beside Methuselah. (This is the type of motion shown in the figure.) Show that the total time lapse as measured by Methuselah is

Fig. 2.8a

(c) Show that in the geometrized units used here, Florence’s acceleration (equal to acceleration of gravity at the surface of Earth) is g = 1.033/yr. Plot T_Methuselah as a function of T_Florence, and from your plot estimate T_Florence if T_Methuselah is the age of the Universe, 14 billion years.

Transcribed Image Text:

. |a| = |a| = √ā. ā.