Planet Earth

Einstein’s Theory of Special Relativity

Classical Physics Review

• Length and Time– Define a standard for each and compare the

unknowns to them.

• Kinematics– derive from length and time velocity and

acceleration– v = x/ t– a = v/ t

• If a is constant– v = at and x = 1/2 a t2

– as t goes to infinity v goes to infinity

Dynamics

• Newton’s Laws (existence and strength of all forces– Law of inertia– F = ma– Law of mutual interaction

• Conservation laws– Momentum– Energy

• Electricity and Magnetism– Maxwell’s equations for the dynamics of charged

particles

Implicit Assumption

• All the laws of nature have the same form in all frames of reference moving at constant velocity with respect to each other.

• Transformation Laws between frames.

S S0vt

x

X0

X0 = x – v t and V0 = V - v

Trouble

• All observers can use the same laws, and communicate!

• Michelson-Morley Experiment– to study the nature of light - particle or wave– if it is a wave - what is the nature of the medium?– search for the ether– the experiment was null

Einstein’s Postulates

• All the laws of nature have the same form in all frames of reference moving at constant velocity with respect to each other.

• The speed of light is a constant for all observers moving at constant velocity with respect to each other.

New Transformation Equations

t = t0 / (1 - v2/c2)1/2

• x = x0 (1 - v2/c2)1/2

• m = m0 / (1 - v2/c2)1/2

• V’x = (Vx - v) / (1 - Vx v/c2)

Time dilation t = t0 / (1 - v2/c2)1/2

• A space ship goes by at v = .8c. Someone on the ship drops a ball 16 ft. and measures a time of flight of 1 sec ( t0 ) . What does a stationary observer measure ( t)?

t = t0 / (1 - v2/c2)1/2 = 1/(1 - .64)1/2 = 1.67 sec.

•

16 ft.

Length Contraction x = x0 (1 - v2/c2)1/2

• If the ball on the spaceship has a diameter of 5 cm ( x0 )as measured by someone on the ship. What is its diameter as measured by a stationary observer ( x)?

x = x0 (1 - v2/c2)1/2 = 5 (1 - .64)1/2 = 5 .6 = 3cm

16 ft.

Mass Increasem = m0 /(1 - v2/c2)1/2

• Same ball has a mass of 10 gm as measured by someone on the ship(m0). What is the mass as measured by the stationary observer (m)?

• m = m0 /(1 - v2/c2)1/2 = 10/(1 - .64)1/2 = 10/ .6 = 16.7 gm

16 ft.

Addition of velocityV’x = (Vx - v) / (1 - Vx v/c2)

• A space ship goes by at .8c (v) and watches

someone on earth turn on a flashlight. What is the speed of the light as measured by the observer on the ship? Note the result is independent of v!

• V’x = (Vx - v) / (1 - Vx v/c2) = ( c - .8c)/(1 - c .8c/c2) = c

S S’ V = .8c

Compare fictitious space ship speed with actual escape speed

• Fictitious speed of .8 c gives the results on the previous slides.

• Actual escape speed is about 7 mi/sec. The speed of light is 186,000 mi/sec.

• The escape speed can be written as .00004c

• try to redo the previous slides with this speed instead of .8c!

• For everyday experience v/c is very small and Einstein’s equations reduce to the ones on slide 4

Implications of variable mass

• The faster an object moves the harder it is to move it any faster.

• Kinetic energy is not the total energy of a moving object– E = KE + m0 c2 = m c2

– Rest energy exist. A particle has energy by virtue of its existence. We have already discussed many applications of this fact!

Suppose the ultimate speed was 100 miles/hour

• What would the three equations predict for a speed of 80 miles/hour or .8c?

• The same as already predicted but you would see it!

Trends

Graph of ratios of masses times and lengths