Addendum 9: Special relativity: Time dilation and length contraction.

First define some constants and dimensional units needed below

1. Time dilation. The constancy of the speed of light for all observers leads to time dilation (and length contraction) of moving observers' time as seen by stationary observers. To compute this, consider the example given below (and in the lecture) of two observers sending light pulses to a mirror at distance w. The stationary observer measures a round-trip time:

Stationary observer (a) half round trip time (and B's half round trip time in his own frame!)

The stationary observer sees the (half) distance traversed by the moving observer's light beam as

Pythagorian theorem

or, solving for t_a

half roundtrip time of observer b's light beam as viewed by observer a

So the time difference is

where the Lorentz factor is defined as:

Example 1: Calculate the time difference among three clocks which are initially synchronized to the same time. One clock remains at the airport, the second flies westward on an around the world flight, while the third flies eastward around the world flight. The planes travel at 800 km/h. The flights take about 50 hours to complete.

Pick center of Earth as 'inertial' stationary frame:

Lorentz time dilation factors for all three clocks wr.t. center of Earth

Time difference between of eastward travelling clock and Earth-bound clock

Time difference between of westward travelling clock and Earth-bound clock

Time difference between of eastward travelling clock and westbound-bound clock

2. Black hole tidal force. The difference in the force of gravity exerted by a body of mass M on one end of a body of mass m to the other (oriented along the radial direction, dimension r) is

Since R >> r, we can use the very useful approximation (obtained from a Taylor series expansion):

(e<<1)

This results in (factoring R, and with e = r/2R, n = -2)

Example: Suppose an astronaut (height 2m) approachs within 1,000 km of a black hole of 3 solar masses. What tidal accleration would she experience? (express in terms of 'g' - the Earth's accerlation). (Hint: Note that acceleration is force per unit mass, so the astronaut's mass drops out.)

This would be immediately fatal!!! Test pilots (e.g. picture) experiencing >10g acceleration black out because the brain no longer receives oxygen from the heart (the heart doesn't generate enough pressure to overcome the 10g accleration).

3. Bending of light (consequence of Principle of Equivalence). Consider a laser beam aimed horizontally in a uniformly accelerating rocket in deep space with acceleration g. According to the Principle of Equivalence, the trajectory is identical to a stationary lab on the Earth's surface with downward graviational acceleration g. For the accelerating rocket, the time taken for a photon to tranverse the horizontal dimension Dx is

In this time the accelerating rocket has moved upward a distance Dy given by

Combining these:

But the acceleration g is

So

or, using the definition of the event horizon above:

Taking Dx to infinity, the bending angle, which we can assume to be small, is approximately

The last term in parantheses is approximately 1, since the bending mostly takes place in the vicinity of the massive object (this an be proven exactly by integration), so we finally have the simple result:

Example: What is the expected angular displacement of a star near the edge of the Sun? Express in arcseconds

This measurement, done during an eclipse of the Sun in 1922 by Sir Arthur Eddington, was the 'classic' test of Einstein's theory of General Relativity. The successful confirmation of the starlight deflection, and hence GR theory, made Einstein instantly famous around the world. (Note, however that this result depends only on the Equivalence Principle, not the full GR theory)

Here are some additional measurements of the starlight defllection experiment during solar eclipses: