At t = 0, an alien spaceship passes by the earth: let this be event A. At t = 13

Question

At t = 0, an alien spaceship passes by the earth: let this be event A. At t = 13 min (according to synchronized clocks on earth and Mars) the spaceship passes by Mars, which is 5 light-minutes from earth at the time: let this be event B. Radar tracking indicates that the spaceship moves at a constant velocity between earth and Mars. Just after the ship passes earth, people on earth launch a probe whose purpose is to catch up with and investigate the spaceship. This probe accelerates away from earth, moving slowly at first, but moves faster and faster as time passes, eventually catching up with and passing the alien ship just as it passes Mars. In all parts of this problem, you can ignore the effects of gravity and the relative motion of the earth and Mars and treat earth and Marsas if they were both at rest in the inertial reference frame of the solar system. Also assume that both the probe and the alien spacecraft carry clocks.
a.) Draw a quantitatively accurate spacetime diagram of the situation, including labeled worldlines for the earth, Mars, and the alien spacecraft, and the probe. Also label events A and B.
b.) Whose clocks measure coordinate times between events A and B? Explain carefully.
c.) Whose clocks measure proper times between these events? Explain.
d.) Does any clock in this problem measure the spacetime interval between the events? If so, which one and why? if not, why not?

Explanation / Answer

A) Point A is at T=0, X=0

Point B is at T=13, X=5

Alien craft moves in a straight line between A and B

Probe moves in a curve between A and B, it starts off vertical at A (at rest wrt earth) and accelerates (bends more horizontal as it increases velocity) and eventually intersects point B

Earth and mars are at rest (vertical lines) at T=0 and T=5 respectively

B) The Earth and Mars clocks (assuming they are synchronized) measure the coordinate times in this frame because they are at rest with respect to (wrt) this frame

C) Both the clocks on the alien spacecraft and the probe.

Since the alien spacecraft is not accelerating, the proper time read on the alien clock is also the spacetime interval

However since the probe is accelerating the time read by the probe clock (its proper time) is no longer the spacetime interval (shorter actually). The two proper times (probe and alien craft) are no longer the same.

D) The clock on the alien spacecraft measures spacetime interval. Since dx is zero for this clock (the alien clock is in the same frame as the alien spacecraft and the alien craft is not accelerating, the interval ds is the same as the interval dt in this frame)

The probe clock does not measure spacetime interval because it is accelerating

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.