Epiphenomena of relativity moving clocks run slow moving

“Epiphenomena” of relativity • moving clocks run slow • moving rulers shrink (in the direction of motion) • moving masses get more massive But motion is relative! If you are moving relative to me, your meter stick is shorter than one meter, and is shorter than my meter stick. But since I am moving relative to you, my meter stick is shorter than one meter, and is shorter than your meter stick. You cannot have A<B and B<A.

The right geometry can tell us about physics A hunter leaves his camp, walks five miles west, and then five miles north. He shoots a bear, and then drags it five miles south back to his camp. What color was the bear? a) b) c) d) e) Brown White Black Pink Puce

The laws of physics are the same in any inertial (nonaccelerating) frame of reference Galileo & Einstein would both agree (at terrestrial speeds. ) F=ma is true, regardless of whether you are on a train, or stationary. x’ = x - v. F t, take two derivatives. The speed of light is independent of the speed of the observer or source. Galileo would say that this is impossible. If you are on a train and throw a ball forward with speed v, then on the ground the ball moves with speed v + vtrain. It’s crazy to suggest that it could move at speed v when measured by me, relative to the train, and also by you, when measured relative to the ground. It may be crazy, but it is true, if the ball is moving at speed c. This is an experimental result, repeated tens of thousands of times.

Note: If by “laws of physics” you mean to include Maxwell’s equations, then postulate 2 is automatically included, since M. E. s predict the speed of light. (Of course, Galileo (d. 1642) was woefully unfamiliar with Maxwell. (b. 1831)

A reference frame is a coordinate system, plus synchronized clocks located at all points in space. An inertial frame is one that is not accelerating.

A reference frame is a coordinate system, plus synchronized clocks located at all points in space. An inertial frame is one that is not accelerating. Acceleration gives “fictitious forces” in the frame. Which of the following is not a fictitious force? Choose any correct answer. A] centrifugal force B] coriolis force C] gravity D] electric force

In an inertial frame, no force still means no acceleration.

Spacetime diagrams & Worldlines What is the worldline of a particle that is accelerating (+ direction)? Which worldline is (that of a stationary particle in) an inertial reference frame?

A train is moving at constant speed. It is shown as the primed frame; the earth is unprimed. Times are in seconds, distances in meters. How fast is the train moving, in m/s? A] 0. 2 B] 0. 5 C] 1 D] 2 E] 5

A little boy throws a ball on the train. The world line of the ball is shown. How fast does the ball move with respect to the earth (in m/s)? A] 0. 2 B] 0. 5 C] 1 D] 2 E] 5 ball

A little boy throws a ball on the train. The world line of the ball is shown. How fast does the ball move with respect to the train (in m/s)? A] 0. 2 B] 0. 6 C] 1. 5 D] 2 E] 5 ball

t=t’ These are vector equations, though this example is 1 D ball

An “event” is something that happens at a point in space and at an instant in time. For example, the child throwing the ball can be thought of as an event. The ball hitting the conductor is another event. e 2 ball e 1

Invariant - a quantity that is the same for all inertial observers. In Galilean relativity, what is invariant? Choose any correct answer. A] the distance between two events B] the time between two events C] the kinetic energy transferred in a collision D] the mass of a particle

These clickers aren’t properly registered. Let me know who you are.

Galilean velocity addition & homework 13. Suppose light moves through ether with speed c and the earth moves through the ether with speed v. If a beam of light moves at an angle ’ wrt to v in the earth frame, how fast does it move?

Michelson Morley Experiment The speed of light is independent of the motion of the earth. It has also been performed with starlight, so “ballistic” theories are ruled out. Light does not move faster if the source is moving! How long does it take to swim across a river & back, vs. upstream and downstream?

Let’s figure out what takes longer: a swim upstream and back downstream, or a swim across and back. What takes longer? a) Up and down b) Across and back c) They take the same time

Michelson Morley Experiment Interferometer. Remember the frequency of light is fixed. If we find out how long light takes to swim up and down, vs. across and back, we can find the phase difference between the beams at A.

Now let’s get back to spacetime diagrams. • We now know the speed of light is the same in any inertial reference frame. • Let’s scale our x and t axes so that a light beam moves with a 45° world line. We will write ct on the t axis. • We can label the x axis in light-seconds (a distance) and the t axis in seconds, with the understanding that we will multiply t by c.

In a spacetime diagram, the x’-axis is always: a) perpendicular to the t’ axis at t’=0 b) the collection of all events that happen at t’=0 c) both d) neither

Spacetime diagrams : revisited. What is the x-axis? It is the set of all events that happen at t=0. These are simultaneous. Any line parallel to the x-axis represents the set of all events happening at time t.

Consider a long spaceship moving in the +x direction. When the middle of the spaceship is at the origin, a light flashes. When does the flash reach the back and the front of the ship? c c WRONG X’

The speed of light is independent of the speed of the observer. So, according to the spaceman, the light reaches the front and the back of his ship at the same time. c c Right A and B are simultaneous to the spaceman. Thus, the line AB is parallel to the x’ axis!

Events that are simultaneous to one observer are NOT to a different observer. c c Right A and B are simultaneous to the spaceman. Thus, the line AB is parallel to the x’ axis!

Note well: you do NOT “drop a perpendicular” to find a coordinate value.

The primed reference frame could best represent: a) a spaceship moving toward +x b) a spaceship moving toward -x c) a stationary spaceship

Two firecrackers explode at exactly the same time, as seen in the x-ct (“lab” or “earth”) frame. What is true in the spaceship frame? A) A explodes first B) B explodes first C) they are simultaneous

Monday A train moving at c/2 turns on its headlight. How fast does the beam of light move forward, according to the engineer? a) c/2 b) c c) 3 c/2 d) 2 c e) Other value

Monday A train moving at c/2 turns on its headlight. How fast does the same beam of light move forward, according to the a person on the ground? a) c/2 b) c c) 3 c/2 d) 2 c e) Other value

The constancy of the speed of light means we must abandon the idea of simultaneity of distant events. What is simultaneous to one observer is not to a different observer. In our spacetime diagrams, this means the x’ axis tilts. The tilt of the x’ axis is the same as the ct’ axis (which is the worldline of the primed frame. )

Two firecrackers explode at exactly the same time, as seen in the x-ct (“lab” or “earth”) frame. What is true in the spaceship frame? A) A explodes first B) B explodes first C) they are simultaneous

Two firecrackers explode at exactly the same time, as seen in the x-ct (“lab” or “earth”) frame. What would be observed by a spaceship moving toward -x? A) A explodes first B) B explodes first C) they are simultaneous

If we arrange that the light from firecracker A causes firecracker B to to explode (without any delay, other than the light travel time), what spacetime diagram shows that? Choose A, B, C, or D -- it would look different in different frames

Slope of the ct’ axis = c/vss. What is the slope of the x’ axis? A) c/vss B) vss/c C) xss/t D) vss/xss

The general form of coordinate transformations

The general form of coordinate transformations Note that every term has units of length

The Lorentz transformation. How do we “calibrate” the axes? I like the light clock. The speed of light is c. Let’s work this on the board. Which is larger, t or t’? a) t b) t’ c) They are the same

Which clock runs slow (in the picture of the light clock we are using)? a) spaceship b) earth c) They are the same

Since the spaceship clock runs slow compared to the earth clock, when the spaceman looks at earth clocks he should see: a) They are running fast, compared to his own b) They are running slow, compared to his own c) They are the same as his own

Why is there no contradiction here? The fundamental entity in relativity is the “event”… something that happens at a specific place and time. Suppose I ask for the time between event A and event B. Observer P says this time is t. P. Observer Q says this time is t. Q. Which is smaller tp or t. Q ? I don’t know. But if tp < t. Q, it is certainly NOT possible that t. Q < t. P !!!!!

What has this got to do with clocks? The events that mark a tick on a light clock are 1) The departure of the pulse from the floor and 2) The return of the pulse to the floor. 1. These happen at the same place in the spaceship, but at different places in the earth frame. 2. What does the spaceman SEE on the earth clocks (that are running slow)? 3. He sees that the earth clocks are NOT PROPERLY SYNCHRONIZED! He says to his earth friend, “of course you think MY clocks are slow… the earth clock that was next to my ship when the light pulse returned to the floor was AHEAD of the clock next to my ship when the pulse first left the floor! 4. Can we see this on the spacetime diagram?

Wednesday - Feb 26 What will different observers disagree about in relativity? Alice and Bob may disagree about when events actually occurred. They figure out when an event (think firecracker explosion) actually occurred by subtracting the light travel time from when they saw the event. e. g. they will disagree about where wavefronts are. Each will observe wavefronts that are equidistant from where the emitter was. In the emitter frame, these are centered on the emitter always. In the lab frame, they are not. What can they NOT disagree about? Alice cannot disagree with Bob about what Bob’s clock says when Bob sees an event. Bob cannot disagree about what Alice’s clock says when Alice sees an event. By the way, there can be no sideways contractions of objects. Why? Pipes!

“Moving clocks run slow”… the light clock. In the spaceship, the clock moves 1 second when the light moves from floor to ceiling and back. In the earth frame, this same light traveled a much greater distance, so the time is longer, by the factor But since the earth is the moving frame, according to the spaceship, earth clocks must run slow, as measured by the spaceship. In the light clock picture on the board, what does the earthframe clock show when the light returns to the floor? a) Since Earth clocks run slow, it will show 1/ b) It will show 1 c) It will show

1. c) It will show 2. It will show a larger elapsed time. 3. BUT!!! This is a different clock from the one used to time the start of the light pulse! Since the spaceship has moved, the pulse returns to the detector in a different place from where it was emitted. 4. So according to the spaceman, the Earth’s clocks run slowly, BUT THEY ARE IMPROPERLY SYNCHRONIZED. Earth clocks that are farther ahead in space (in the direction of the spaceship’s motion) are set too much ahead.

Note that the spaceman IS NOT WRONG. The earth clocks are improperly synchronized. But the earthman IS NOT WRONG. The earth clocks are properly synchronized in the Earth reference frame. ---Now, let’s finish calibrating our axes and deriving the Lorentz transformation.

Lorentz contraction If we measure a time interval between two events at the same place in the spaceship (light the light clock), we get a bigger interval than the spaceship clock shows. If we measure a space interval (length of a meter stick), we get a smaller interval than the spaceship gets.

What is the length of a meter stick (if it’s moving)? Answer: it is the distance between the ends, measured at the same time!

The spacetime interval is invariant. For half a light clock cycle, this interval is L, which is manifestly invariant

Spacelike separated events are simultaneous in some frame. They occur in the opposite order in some frames.

If a particle has velocity vx in the unprimed frame, what is its velocity in the primed frame? Just take differentials of Lorentz. Inverse transform has V -> -V

Lengths in y, z don’t change, but velocities do. Why?