Special Theory of Relativity

Special Relativity I• Einstein’s postulates

• Simultaneity

• Time dilation

• Length contraction

• New velocity addition law

I: EINSTEIN’S POSTULATES OF RELATIVITY

• Postulate 1 – The laws of nature are the same in all inertial frames of reference

• Postulate 2 – The speed of light in a vacuum is the same in all inertial frames of reference.

• Let’s start to think about the consequences of these postulates.

• We will perform “thought experiments” (Gedankenexperiment)…

• For now, we will ignore effect of gravity – we suppose we are performing these experiments in the middle of deep space

I. Invariance of the speed of light

II. Simultaneity

III: TIME DILATIONA light clock consists of two parallel mirrors and a photon bouncing back and forth over the distance D. An observerat rest with the clock will measure a click at times

to= 2H/c

• Now suppose we put one clock on a train or spaceship that is cruising (at constant velocity, v) past us.

• How long will it take the clock to “tick” when its in the moving spacecraft? Use Einstein’s postulates…

Now suppose that we put the clock on a platform sliding at constantspeed v.

vt

H d

d2=H2+(vt/2)2

• Clock appears to run more slowly.• But, suppose there’s an astronaut in the spacecraft

– the inside of the spacecraft is also an inertial frame of reference – Einstein’s postulates apply…

– So, the astronaut will measure a “tick” that lasts

• So, different observers see the clock going at different speeds!

• Time is not absolute!

to=H/c

• Effect called Time Dilation.• Clock slows by a factor of

• This is called the Lorentz factor, or

22

22

/cv-1

1 /1

c

Hcvc

H

22/cv-1

1

Lorentz factor

Examples of time dilation

[We will work through these examples on the white board during the class]

• Fast moving spacecraft• The Apollo mission to the Moon• Clocks flown in airliners• Normal everyday life• The Muon Experiment• The jet in the galaxy M87

The jet in M87

IV: LENGTH CONTRACTION

• Consider two “markers” in space.

• Suppose spacecraft flies between two markers at velocity v.

• Compare what would be seen by observer at rest w.r.t. markers, and an astronaut in the spacecraft…

• So, moving observers see that objects contract in the direction of motion.

• Length contraction… also called– Lorentz contraction

– FitzGerald contraction

• Consider Muon experiment again, this time from point of view of the Muons– i.e. think in frame of reference in which Muon is at rest

– Decay time in this frame is 2 s (2/1000,000 s)

– How to they get from top to bottom of Mountain before decaying?

– From point of view of Muon, Mountain’s height contracts by factor of

– Muons can then travel reduced distance (at almost speed of light) before decaying.

New velocity addition law

• Once we’ve taken into account the way that time and distances change, what’s the new law for adding velocities?

221

21

1 cVV

VVVadd