Relativity History 1879 Born in Ulm Germany 1901

Relativity: History • 1879: Born in Ulm, Germany. • 1901: Worked at Swiss patent office. – Unable to obtain an academic position. • 1905: Published 4 famous papers. – – Paper on photoelectric effect (Nobel prize). Paper on Brownian motion. 2 papers on Special Relativity. Only 26 years old at the time!! • 1915: General Theory of Relativity published. • 1933: Einstein left Nazi-occupied Germany. – Spent remainder of time at Institute of Advanced Study in Princeton, NJ. – Attempted to develop unified theory of gravity and electromagnetism (unsuccessful).

Relativity

Thought Experiment - Gedanken The Special Theory of Relativity Einstein asked the question “What would happed if I rode a light beam? ” • Would see static electric and magnetic fields with no understandable source. • Electromagnetic radiation requires changing E and B fields. • Einstein concluded that: • No one could travel at speed of light. • No one could be in frame where speed of light was anything other than c. • No absolute reference frame

Einstein’s Postulates or • All inertial frames of reference are equivalent with respect to the laws of physics • No experiment one can perform in a uniformly moving system in order to tell whether one is at rest or in a state of uniform motion. (No dependence on absolute velocity. ) • The speed of light in a vacuum always has the same value c, independent of the motion of the source or observer. or • Nothing can move faster than the speed of light in a vacuum, which is the same with respect to all inertial frames

Space-Time Diagram Requirement for 4 Dimensions Definitions: • Event: characterized by location (e. g. , x, y, z) and time (t) at that location • Space-time diagram: a coordinate system in which every point represents an event. 4 dimensions required. • World line: trajectory of an event in the space-time diagram ct World line A (ct, x) O x

Description of Motion In a spacetime diagram, the motion of an object traces out a world line. For an object that moves at a constant velocity, a simple way of measuring the velocity is to measure the positions of the object at two different times. Assume that the object moves from r 1 at t 1 to r 2 at t 2, the velocity of the object is then We need a way of synchronizing the clocks at different locations!

Synchronization of Clocks According the Einstein’s second postulate, no information can be transmitted at a rate greater than the speed of light in vacuum. Since the speed of light is independent of inertial frames, it provides a natural (and ideal) way of sychronizing clocks. The procedure can be described as follows: • Choose a reference clock and reset it to zero • Generate a light pulse from the location of the reference clock • Set a local clock to the time that it takes for the light pulse to propagate from the location of the reference clock to the current location.

Time Intervals: Simultaneous Events • Two events simultaneous in one reference frame are not simultaneous in any other inertial frame moving relative to the first. Two lightning bolts strike A, B Right bolt seen first at C’ Two bolts seen simultaneously at C Left bolt seen second at C’

Relativity of Simultaneity Two events simultaneous in one inertial frame are not simultaneous in any other inertial frame moving relative to the first OR Clocks synchronized in one inertial frame are not synchronized in any other inertial frame moving relative to the first ct ct ct’ x’ x O A B C

Light Clock • Light pulse bouncing between two mirrors perpendicular to direction of possible motion • A one way trip is one unit of time t = d/c • Clearly moving light clock has longer interval between light round trips

Handy Light Clock Consider pulse of light bouncing between two mirrors (retroreflectors) d to = d / c

Now Observe Same Clock moving Thought Experiment Gedanken Experiment Consider an inertial frame of reference: Elevator moving upward at a constant velocity, v.

Moving Light Clock Consider path of pulse of light in moving frame of reference: Light Clock ct d vt to = d / c

Time Dilation calculated Use Pythagorean Theorem: ct (ct) 2 = d 2 + (vt) 2 d d 2 = (ct) 2 - (vt) 2 d 2/ c 2 = t 2 - (v 2/ c 2)t 2 d / c = t [1 - (v 2/ c 2)]1/2 But d = cto , So to = t [1 - (v 2/ c 2)] 1/2 The clock in the moving frame runs slower. vt

Time Dilation Observed! Does this really work? to = t [1 - (v 2/ c 2)] 1/2 t = gt o 1. Mu-Mesons last longer before decaying if they are moving very fast. 2. by factor g = 1/ [1 - (v 2/ c 2)] 1/2 2. Atomic Clocks run slower when moving. 3. 1 sec/1 000 sec at 675 mph.

Time Dilation: Derivation Analyze laser “beam-bounce” in two reference frames In S’ frame, light travels up or down a distance D. In S frame, light travels a longer path along hypotenuse. • Solve for t • Substitute t’ = D/c (proper time)

Time Dilation/Length Contraction: Muon Decay • Why do we observe muons created in the upper atmosphere on earth? Muon’s frame Length Contraction Earth’s frame Proper lifetime is only = 2. 2 s travel only ~650 m at 0. 99 c • Need relativity to explain! Time Dilation – Time Dilation: We see muon’s lifetime as = 16 s. – Length Contraction: Muon sees shorter length (by = 7. 1)

Length Contraction • Necessary consequence of postulates and for consistency of effects • Can also derive in four dim. (ct, x, y, z) as rotation in a space-time plane preserving 4 -D length, like rotation in a space-space plane preserve length 3 -D 4 -D Pythagorean Theorem

Relationship between Inertial Frames ct ct’ x’ O O x

Light Cone Unchanged • If the speed of light is identical for all inertial frame observers, then the light cone must be unchanged.

Aberration of Light • Discovered by Bradley in 1725 after seeing pennant on sailboat having direction intermediate to wind and boat motion.

Doppler Effect

Relativity Warp 0. 92 (0. 75 c)

Relativistic Increase in Mass • E = m 0 c 2 = m 0 c 2 E • m = m 0 E = m c 2 v v=c

Energy & Momentum 3 -D Case 4 -D Momentum Energy and Momentum are separate in 3 -D and have separate conservation laws. In 4 -D are part of same vector and rotations preserve length (norm).

Rest Mass • The rest mass m 0 of a particle is an invariant. It is the length of the 4 -D momentum vector.

Einstein’s General Theory of Relativity predicts black holes • Mass warps space resulting in light traveling in curved paths

Principle of Equivalence A homogeneous gravitational field is completely equivalent to a uniformly accelerated reference frame. It is impossible for us to speak of the absolute acceleration of the system of reference, just as theory of special relativity forbids us to talk of the absolute velocity of a system.

Equivalence Principle Consider an observer in an elevator, in two situations: mi = m g 1) Elevator is in free-fall. Although the Earth is exerting gravitational pull, the elevator is accelerating so that the internal system appears inertial! 2) Elevator is accelerating upward. The observer cannot tell the difference between gravity and a mechanical acceleration in deep space!

Uniformly Accelerating Frame Light in Accelerating Frame of Reference Gravity? acceleration

Time Dilation in Gravitational Field • Clock lower down runs slower

Is it General Relativity right? • The orbit of Mercury is explained by Relativity better than Kepler’s laws • Light is measurably deflected by the Sun’s gravitational curving of spacetime. • Extremely accurate clocks run more slowly when being flown in aircraft & GPS satellites • Some stars have spectra that have been gravitationally redshifted.

If we apply General Relativity to a collapsing stellar core, we find that it can be sufficiently dense to trap light in its gravity.

Several binary star systems contain black holes as evidenced by X-rays emitted

Cygnus X -1 must have a mass of about 7 times that of the Sun

Other black hole candidates include: • LMC X-3 in the Large Magallenic Cloud orbits companion every 1. 7 days and might be about 6 solar masses • Monoceros A 0620 -00 orbits an X-ray source every 7 hours and 45 minutes and might be more than 9 solar masses. • V 404 Cygnus has an orbital period of 6. 47 days which causes Doppler shifts to vary more than 400 km/s. It is at least 6 solar masses.

Supermassive black holes exist at the centers of most galaxies

Supermassive black holes exist at the centers of most galaxies

Primordial black holes may have formed in the early universe • The Big Bang from which the universe emerged might have been chaotic and powerful enough to have compressed tiny knots of matter into primordial black holes • Their masses could range from a few grams to more massive than planet Earth • These have never been observed • Mathematical models suggest that these might evaporate over time.

How big is a black hole?

Matter in a black hole becomes much simpler than elsewhere in the universe • No electrons, protons, or neutrons • Event horizon – the shell from within light cannot escape • Schwarzschild radius (RSch) – the distance from the center to the event horizon • gravitational waves – ripples in spacetime which carry energy away from the black hole • The only three properties of a black hole – mass, angular momentum, and electrical charge

Structure of Schwarzschild Black Hole

Structure of Kerr (Rotating Black hole Structure of a Kerr (Rotating) Black Hole In the Erogoregion, nothing can remain at rest as spacetime here is being pulled around the black hole

Falling into a black hole is an infinite voyage as gravitational tidal forces pull spacetime in such a way that time becomes infinitely long

Black Hole Evaporation: Caused by virtual particles

Black holes evaporate Virtual particles that appear in pairs near a event horizon may not be able to mutually annihilate each other if only one manages to survive a trip along the event horizon.

Summary • Special Relativity yields: – Lost of universal simultaneity – Time dilation of moving systems – Length Contraction of moving objects – Equivalence of Mass and Energy – Integrated 4 -Dimensional space-time • General Relativity / Equivalence Principle – Curved Space-Time – Time Dilation in gravitational potential (curved time) – Bending of light and all inertial paths (no gravity) – Black Holes – Matter/Energy tells spacetime how to curve, spacetime tells matter/energy how to move

Appendix: 4 -D Vectors • Summary discussion of four dimensional (4 -D) vectors • Have vector algebra just like 3 -D vectors but have 4 components instead of 3: Transformations that leave length unchanged are the familiar: 1) Translations - displacements in space or time 2) Rotations - angular rotations 3) Velocity boosts (Lorentz transformation) which are equivalent to rotations in a space-time plane

Lorentz Transformations Leave 4 -D vectors length Invariant

Examples of 4 -D Vectors Easiest way to see that 4 -D vector transforms like the prototype under Lorentz transformations is to construct them that way!

Velocity Composition 3 -D Velocity 4 -D Velocity Lorentz Transformation Change of velocity is simply a rotation through and angle