Relativity : Revolution in Physics

Einstein’s Triumph

Special Relativity - 1905

• How motion in space is related to motion in time

• Applies to “inertial frames,” moving with respect to one another at constant velocity

• Does not apply to accelerated motion• Requires changing Newton’s Three Laws

of Motion for high velocities• Requires changing definitions of

momentum and energy

Strange Predictions

• Stretching (“dilation”) of time

• Contraction of length• Existence of rest

mass according to E = mc2

Basics

• All motion is relative – must be defined with respect to a particular “frame” of reference. Examples:– The ground– A moving bus

• Q: What is speed of a 30 m/s baseball relative to you thrown from a truck traveling at 20 m/s toward or away from you?

Two Postulates

• The laws of physics are the same in any inertial (that is, non-accelerated) frame of reference. – This means that the laws of physics observed by a

hypothetical observer traveling with a relativistic particle must be the same as those observed by an observer who is stationary in the laboratory

• The speed of light is the same for all observers, no matter what their relative speeds.

First Postulate

• All the laws of nature are the same in all uniformly moving frames of reference

• Consequences:– All frames of reference are arbitrary– Absolute motion does not exist– No possible experiment confined to a vehicle

can detect its motion (you can detect motion by looking outside)

Second Postulate – The Speed of Light is Constant, Regardless of the

Motion of Source or Observer

• Light does not behave like the baseball!

• Imagine a flashlight beam directed from a spaceship moving with nearly the speed of light. All measurements of the speed of this light give same answer; 300,000 kilometers per second

Courtesy www.cybersurfari.org/images/promos/ fall2k2/spaceship.gif

Thought Experiments

• Einstein had deep insight into nature

• Remarkable powers of concentration

• Great ability to visualize events

• YOU can improve your powers too

• Excellent Einstein link: http://whyfiles.org/052einstein/genius.html

Courtesy of the Archives, California Institute of Technology

Einstein’s Thought Experiments

• Imagined riding alongside beam of light; concluded it was impossible

• Train experiment. Made shocking discovery that simultaneity of events depended of motion of the observer

• Light clock – discovered time dilation and how big it is

• Link to Einstein’s Thought Experiments http://aether.lbl.gov/www/classes/p139/exp/gedanken.html

Train Experiment

Courtesy homepage.mac.com/ardeshir/ TrainImage.jpg

Light from each lightning strike does not arrive at observer’s position at same time; delays depend on speed of train

Light Clock Experiment

Light clock moving

Light clock at rest

Light Clock Experiment

ct

vt

ct0

Light path as seen by observer at rest

c2t2 = c2t02 + v2t2

Shows three positions of light clock as it moves to the right at speed v. By Pythagorean theorem

Zero subscript refers to non-moving frame

Time Dilation Equation

c2t2 = c2t02 + v2t2

c2t2 - v2t2 = c2t02

t2[1 – (v2/c2)] = t02

t2 = t02 /[1 – (v2/c2)]

t = t0 [1 – (v2/c2)]-1/2

Interpretation

• Compared to a system at relative rest, time passes more slowly in a moving system

• Within a given system (rest frame) no relativistic effects are noticed

• Example: 30 minute waste basket fire on a spaceship traveling at v = 0.8 c. How long will this fire appear to last when seen from Earth?

Wastebasket Fire

t0 = 30 min

v/c = 0.8

t = t0 [1 – (v2/c2)]-1/2

t = 30min[1 - 0.64]-1/2

t = 30/0.6

t = 50 minutes

Variation With v/c

• Complete this table

v/c (v/c)2 1 – v2/c2 [1 – v2/c2] 1/2

1

0.8

0.9

0.99

Variation With v/c

• Complete this table

v/c (v/c)2 1 – v2/c2 [1 – v2/c2] 1/2

1 1 0 0

0.8 0.64 0.36 0.6

0.9

0.99

Variation With v/c

• Complete this table

v/c (v/c)2 1 – v2/c2 [1 – v2/c2] 1/2

1 1 0 0

0.8 0.64 0.36 0.6

0.9 .81 .19 0.43

0.99

Variation With v/c

• Complete this table

v/c (v/c)2 1 – v2/c2 [1 – v2/c2] 1/2

1 1 0 0

0.8 0.64 0.36 0.6

0.9 .81 .19 0.43

0.99 .98 .02 0.14

Twin Trip

• According to time dilation a twin astronaut on a high speed trip returns younger than his/her twin because time runs more slowly for the moving twin compared to the stay-at-home.

• If the trips lasts 30 years at v/c = 0.8 and both twins are initially 20 years old, what will be their ages on earth afterward?

Paradox

• Since motion is relative why couldn’t we look at the trip from the point of view of the traveling twin (who sees the earth recede when he/she leaves). Then wouldn’t the traveling twin age more?

Courtesy University of New South Wales, Australia

Resolution of Paradox

1. Argument in Hewitt Chapter 15 section 7.

2. Trip involves acceleration which special relativity does not include. How many times must traveling twin accelerate or decelerate (if all motion is straight line) before returning? In other words problem is not symmetrical because traveling twin has several reference frames while stay at home has only one

Using General Relativity, which does include acceleration it is also possible to show that stay at home twin ages more

Twin Paradox Resolution, con’t

3. World Line in spacetime argument http://physics.syr.edu/courses/modules/LIGHTCONE/twins.html

Link to multiple explanations:

http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html

Implications for Space and Time Travel

• Very long space trips are possible within a single lifetime

• Require speeds comparable to c• For v/c = 0.999 70 years on Earth elapse in only

one year of a travelers time.• Traveler could journey to a star 35 ly from earth

and return in only one year of travelers time.• On longer trips would return to a different

century• A form of time travel – into future

• Viewed from frame of reference of light how much time is required for a journey to the center of our galaxy, 30,000 ly away?

• Answer: None. In frame of reference of a light wave time stands still.

Brain Teaser

Length Contraction

• Moving objects contract along their direction of motion (not perpendicular)

• L = L0 [1 – v2/c2]

• What would be the length an Earth observer would measure for a spaceship traveling at v/c =0.8 if an observer on the ship measures 100 meters?

• Answer: 60 meters

Puzzler

• An observer on earth measures the length of a rocket ship traveling at v = 0.8c to be 60 meters. What would be the measured length of this ship when at rest on Earth?

• Answer 100 meters

Mass and Energy

• Mass is a form of energy• Even object at rest has energy• Rest energy is E0 = mc2

• All exo-energetic (energy producing) reactions, chemical and nuclear, get their energy from mass. The reaction products have less mass than the reactants!

• In chemical burning the difference is less than one part in a billion

• In nuclear fission about one part in a thousand

Rest Energy of a Kilogram

• Problem: Calculate the rest energy in joules of one Kilogram

• E0 = mc2

• E0 = 1Kg (3.0 x 108 m/s)2

• E0 = 9 x 1016 Joules

Amount Available from Burning

• Take the ratio of one in a billion joules available from chemical burning and apply to E = 1017 Joules to find the amount available from burning 1 kg of coal.

• Answer; Approx. 108 J

• Is this reasonable? Make an argument or do some research to find out!

Relativistic Momentum

• Replace mv by

• p = mv [1 – (v2/c2)]-1/2

• As speed approaches speed of light, what happens to p?

• Answer: approaches infinity

• How much force x time would be required to accelerate it further?

• Answer: infinite

Brain Tickler

• Based on relativistic increase in momentum why is c the ultimate speed limit in the universe?

• Answer: because infinite force is required to get there and infinite force cannot exist

Alternate Interpretation of Relativistic Momentum

• Some physicists write p = mv where

• m = m0 [1 – (v2/c2)]-1/2

• This is called relativistic increase of mass

Relativistic Kinetic Energy

• Instead of KE = ½ mv2

• KE = mc2 [1 – (v2/c2)]-1/2 – mc2

• Challenge: show that the above expression simplifies to ½ mv2 in the limit that v approaches zero.

Correspondence Principle

• Relativity doesn’t replace Newton’s physics, it extends it to high speeds. So all relativistic equations must reduce to Newtons when speeds are low.

• Examples: L = L0 [(1 – v2/c2]

t = t0 [1 – (v2/c2)]-1/2

p = mv [1 – (v2/c2)]-1/2

Unanswered Questions

• What is time?

• Why does it only run forward?

• Could there be universes where it runs backward (anti matter particles can be conceived as particles running backward in time)

• Was time created by the Big Bang or did it exist before?