A particle accelerator is a device that boosts subatomic particles to speeds clo

Question

A particle accelerator is a device that boosts subatomic particles to speeds close to that of light. Such an accelerator is typically shaped like a ring (which may be several kilometers in diameter): the particles are constrained by magnetic fields to travel inside the ring. Imagine such an accelerator having a radius of 2.998 km. Assume there are two synchronized clocks (P and Q) located on opposite sides of the ring. A certain particle in the ring is measured to travel from clock P to clock Q in 34.9 s, as registered by those clocks. Let event A be the particle's departure from clock P and event B be the particle's arrival at clock Q. Assume the particle contains an internal clock that measures the time between these events, and that the particle travels at a constant speed. (a) What is the particle's speed in the laboratory frame? Answer (b) Does the synchronized pair of laboratory clocks measure the proper time, the coordinate time, or the spacetime interval between events A and B? (c) Does the particle's internal clock measure the proper time, the coordinate time, or the spacetime interval between events A and B? (d) What is the spacetime interval between these events?

Explanation / Answer

a)

v= dx/dt = (1/2)2 pi R/dt = (1/2) 2 pi 2.998 km/ 34.9x10-6 sec = 2.7x105km/sec

b)

The synchronized laboratory clocks do not measure proper time between events A and B

The synchronized laboratory clocks do measure coordinate time in the lab frame

The synchronized laboratory clocks do not measure the space time interval between events A and B

Easy way to remember what proper time is: Proper time is what you measure with YOUR wristwatch. e.g. not moving with respect to you and dx = 0 (strapped to your wrist, not 100 miles away)

(The problem with the distant clock is that if you start moving, it will not remain synchronized, thus while a distant clock, if it is not moving with respect to you, could be measuring proper time, a local clock, strapped to your wrist, always will measure proper time)

Another nice thing about proper time: proper time IS the spacetime interval for an inertial reference frame (since dx=0 for your watch in your rest frame), and the spacetime interval is the same for all observers.

The events A and B are separated in space (dx not 0)

The proper time ds given by ds2 = dt2 + dx2

Since dx is not zero, ds does not equal dt

ds (the proper time) does not equal dt (what the clocks are measuring)

The sychronized lab clocks do measure coordinate time in the lab frame (they are synchronized with each other, and not moving with respect to the lab frame).

The synchronized lab clocks do not measure the spacetime interval ds. Note the two events are separated in space.

Since dx is not zero, ds does not equal dt

c)

Particle's internal clock measures the proper time (not to beat this to death, but the particle's clock is in effect a wristwatch strapped to the particle).

Particle's internal clock does not measure the coordinate time in the lab frame. We concluded in part B that the lab clocks are measuring coordinate time in the lab frame which is not proper time (for the particle). The converse of this is that if the particle's clock is measuring proper time it is not measuring coordinate time (in the lab frame).

Particle's internal clock does not measure the spacetime interval. The particle is accelerating in a circle (no an inertial reference frame), so the proper time is no longer the spacetime interval.

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