Your head is about to hurt! Yes Alex even more!!! Relativity Your head is about to hurt! Yes Alex even more!!!

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).

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 All inertial frames of reference are equivalent with respect to the laws of physics or 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 World line ct x O A (ct,x) 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

Description of Motion We need a way of synchronizing the clocks 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 r1 at t1 to r2 at t2, 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 x O A B C x ct O A B C ct’ x’

Light Clock Light pulse bouncing between two mirrors perpendicular to direction of possible motion A one way trip is one unit of time Dt = 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 vt d to = d / c

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

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

Time Dilation/Length Contraction: Muon Decay Why do we observe muons created in the upper atmosphere on earth? Proper lifetime is only  = 2.2 s  travel only ~650 m at 0.99c Need relativity to explain! Time Dilation: We see muon’s lifetime as  = 16 s. Length Contraction: Muon sees shorter length (by g = 7.1) Length Contraction Muon’s frame Earth’s frame Time Dilation

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 Pythagorean Theorem 3-D 4-D

Relationship between Inertial Frames x ct O ct’ x’ O The world line of light always makes the same angle with the spatial axis (x or x’) as with the time axis (t or t’).

Defining Relativity All measurements are “relative” to your velocity. As you speed up (v), length contracts and time dilates according to the gamma (γ) factor.

Doppler Effect

Relativistic Increase in Mass E = gm0c2 = m0c2 m = gm0 v E v = c E = m c2

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 the 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 = mg 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

Is it General Relativity right? 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.

Black hole candidates include: LMC X-3 in the Large Magallenic Cloud orbits its companion every 1.7 days and might be about 6 solar masses Monoceros A0620-00 orbits an X-ray source every 7 hours and 45 minutes and might be more than 9 solar masses. V404 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

Summary Special Relativity yields: Loss 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