Philosophy of Physics III Special Relativity

Special Relativity in two hours SR is conceptually hard, but mathematically easy. All you need is high school algebra. The basics (the kinematical part) can be learned in two hours. Why bother learning it? –Central to physics –Philosophically interesting –Culturally important

Special Relativity (SR) Albert Einstein ( ) “Annus mirabilis” 1905 –Special Relativity (SR) –Brownian motion (reality of atoms) –Light quanta (photons and photoelectric effect)

SR grew out of problems with electrodynamics SR reconciles mechanics and electrodynamics by revising central concepts of space and time. Space and time, formerly separate, become united as spacetime. SR is central to contemporary physics and is philosophically interesting & important.

Postulates of Special Relativity 1.The laws of nature have the same form in every inertial frame. This is often called “the principle of relativity” It was first formulated by Galileo and Descartes. The idea is central to Newton’s first law: A body moves at constant velocity (ie, constant speed in a straight line), unless acted upon by a force. (This is inertial motion) An inertial frame is any coordinate frame moving at constant velocity. Einstein’s claim is that the laws of nature are the same in any inertial frame. This is not new; Galileo and Newton said the same.

2.The velocity of light (in a vacuum) is the same in every frame. The velocity of light is c = 3 x 10 8 m/s It has this value in every frame, regardless of the motion of the source or the receiver of the light. Adding velocities, according to SR, will not give the result we would expect in either Newtonian physics or common sense.

Frames of Reference Also called coordinate frames. There are infinitely many frames, moving in every direction, at every velocity (relative to one another), up to (but not including) the speed of light. For convenience we often talk about the frame of the room or the frame of the rocket, etc. This means, the frame in which the room or the rocket is at rest. A frame extends to infinity in all directions. Every object and every event is in every frame. Thus, I am in the frame of the room and also in the frame of the rocket. I am at rest in the former and moving in the later. But am I really at rest and only appear to be moving? NO. All frames are equivalent. I am really at rest in the room frame AND I am really moving in the rocket frame. Both are true.

Simultaneity Einstein characterized simultaneity within a frame as follows: Two spatially separated events are simultaneous in the inertial frame F iff light signals from the events meet at a point midway between the events. Think of events as points, tiny explosions that emit light.

Exercise 1: Give definitions of earlier and later for pairs of events based on the definition of simultaneity.

Philosophical problem. Don’t worry too much about this problem now, but ask yourself this for next week: Is Einstein’s definition of simultaneity based on anything that might be called evidence, or is it an arbitrary stipulation that happens to be convenient?

Relativity of Simultaneity The definition given above is for simultaneity within a single frame. In classical physics and in common sense, if two events are simultaneous in any frame they are simultaneous in every frame. This is not so in SR.

Suppose we have two frames, the train and the track; the train is moving at velocity v in the track frame. Let e 1 and e 2 be two separated events (flashes) The observer in the track frame is midway between e 1 and e 2, and receives signals from each at the same time. Therefore, e 1 and e 2 are simultaneous in the track frame.

However, some time passes while the light signals come to the mid-points, so the train has moved forward (or the ground has moved back). An observer midway on the train frame receives the signal from e 2 before e 1. So e 2 was earlier than e 1 in the train frame. Therefore: Simultaneity of events is relative to a frame.

Caution Resist the temptation to ask: Where did the flashes really happen?, and expect an answer such as: “in the track frame.” They happened at two points (several metres apart) in the track frame and they happened at two points in the train frame (front and rear of the train). Both descriptions are true and neither frame is the unique “real” one. Always remember the equality of frames.

Time Dilation In classical (Newtonian) physics and common sense, the time interval between two events is the same in all frames. –Eg, the time between two of my heart beats (as measured while at rest in this room) is one second. As measured by a rocket whizzing by it would also be one second. This will not be so in SR. Different frames will give different temporal intervals between events. Definition: The temporal interval between two events which happen at the same spatial location, as measured in a given frame, is the proper time.

Computing Time Dilation Suppose two events are separated by a time interval Δt in frame F. In a frame F′ moving with velocity v relative to F, the temporal interval Δt′ between the pair of events will be lengthened (dilated) by a factor of Thus, in F′ the time interval is

Notice two things about As v gets smaller, Δt′ gets closer to Δt. As v gets bigger (ie, closer to c), Δt′ gets larger (if v=c, then Δt′ is infinite).

Here’s why there is time dilation Suppose a light clock sends successive signals to a mirror. One round trip is one “tick” of the clock. The time it takes light to travel distance l to the mirror is Δt, so the time between ticks of the clock is 2Δt.

Next, suppose we have a pair of synchronized clocks, C 1 and C 2, which are at rest in frame F and a clock C′ at rest in frame F′, which is moving with relative velocity v. The time between e 1 and e 2 in frame F′ is just one tick (round trip), which is: 2Δt′ = 2 l/c (If you were at rest in F′, this is the time you would measure between ticks.) The time (for the round trip) as measured in F using F’s light clock is 2Δt. (If you were at rest in F, this is the time you would measure between ticks.)

The derivation Now we want to relate the time between ticks of the F′ clock, as measured by the F clock, that is, the relation between 2Δt and 2Δt′ (or more simply, between Δt and Δt′), in the different frames. NB. In this case, we use 2Δt to indicate the time of one tick of the clock in the moving frame. This is not the time of one tick of the C clock. It is the time, as measured by the C clocks, of one tick of the C′ clock. From the diagram, using the Pythagorean theorem, we have:

Note: The term is often called the “correction factor.” As mentioned before, it goes to 1 as v goes to 0; and it goes to 0 as v goes to c. It becomes significant only at high velocities. Exercise 2: A rocket is going to Andromeda, which is two light years away. Its velocity is.95c. (a) How long will it take to get there in the earth’s frame?; (b) How long to get there in the rocket’s frame? (To compute your answer, let c=1. and let v=.95. You will find working in “natural units” easier.)

Length contraction Definition: The length of an object in a frame in which it is at rest is called its rest length or its proper length. In classical (Newtonian) physics and common sense, the length of any object is the same in every frame. That is, if a rod is, say, one metre long when at rest in the classroom, then it is also one metre long as measured by a passing rocket. This is not so in SR.

Computing contraction An object with rest length L in frame F, which has velocity v with respect to F′, will have length L′ in F′, where This is known as Lorentz contraction.

Notice two things about As v becomes smaller, L′ approaches length L. As v gets closer to c, L′ approaches 0.

Here’s why contraction Consider a set-up like the one above with a rod that has rest length L. It is situated between two clocks C 1 and C 2 in frame F, which is moving with velocity v with respect to F′. We want to calculate the rod’s length, L′, in frame F′.

Let a light signal from C′ to the mirror be sent when the front of the rod passes the point where C′ is located and return to the point at C′ when the end of the rod passes. (We can adjust the distance l to the mirror so that this works out.) Let this time interval be 2Δt′. Since length = velocity x time, we have L′ = v2Δt′ This is the length of the rod in F′, in which it is moving at velocity v.

In frame F the rod has length L, which is the distance the clock C′ moves in the time 2Δt. Thus, L = v2Δt. Now we can compute L′

Relational Facts In common sense and in classical physics, things are moving or they are not; they have length L or they don’t, and so on. Facts are absolute. But in SR, motion, simultaneity, length, etc. are relative to a frame. Absolute: –X has velocity v (regardless of frame) –X has length L (regardless of frame) Relative: –X has velocity v in frame F –X has velocity v′ in frame F′ –X has velocity v′′ in frame F′′ –And so on.

But not everything is relative. Some things are frame-independent, ie, true in every frame: –X is the mother of Y –Species evolve –2+3=5 –Sunsets are beautiful –Murder is wrong Don’t make the mistake of falling into some kind of moral or epistemic relativism. SR does not imply that.

Exercise 3: What is the distance from Earth to Andromeda in the rocket frame? (See earlier exercise.) Exercise 4: Suppose the rest length of a car is 7 m and the garage length is only 6 m. The garage owner says: “If the car goes fast enough, it will fit inside.” The driver says: “If our relative velocity is high, then the garage will be even shorter, so the car can’t possibly fit in.” Who is right? Resolve this paradox. (Hint: think about simultaneity.)

Velocity addition We won’t derive the velocity addition rule here, but only postulate it. If an object is moving at u in F′ and F is moving at v in F, then the object’s velocity in F is w.

Exercise 5: If a sprinter is moving at 10 m/s on a train that is moving in the same direction at 25 m/s, how fast is the sprinter moving with respect to the track according to SR? Exercise 6: Make up some problems of your own involving a variety of different speeds, from slow to close to c.

Take a breath You have now seen the conceptual basics of SR (kinematics, but not dynamics). Master that and you are doing well. You can stop here, if you want. Of course, there’s lots more to SR, so push on if you want to see some additional interesting stuff.

Lorentz Transformations An event will have coordinates (x,y,z,t) in F and (x′,y′,z′,t′) in F′. We want to know how to change from one set of coordinates to another. By convention, we let the two frames move along the common x-x′ axis and we set t=t′=0, when the origins coincide. As time passes, the distance apart of the origins is vt in F and vt′ in F′.

Lorentz transformation equations It is easy to derive the Lorentz transformation, but here it will just be postulated.

Or, if we’re starting from the primed frame,

Exercise 7: A muon is created in the upper atmosphere and travels down at v =.99c. After 5 Km it decays. (a) What is its lifetime as measured by us? (b)as measured in its own frame? (c)how thick is the atmosphere it travels through as measured in the muon’s frame?

Spacetime intervals Definition: The interval (squared) between two events is: I 2 = Δt 2 – Δx 2 – Δy 2 – Δz 2 (where Δt = t 2 -t 1, etc.) The events e 1 and e 2 have both primed and unprimed coordinates. The interval in unprimed coordinates is: I′ 2 = Δt′ 2 – Δx′ 2 – Δy′ 2 – Δz′ 2 (where Δt′ = t′ 2 - t′ 1, etc.).

Invariance of the Interval Here is a remarkable fact: I 2 = I′ 2 Lengths and times may be frame relative, but a spacetime interval is not. A given interval is the same in every frame. Einstein and others called the theory “Invariance theory” in the early days, but “Relativity theory” stuck (which is perhaps unfortunate).

Exercise 8: Prove a 2-dimensional version of the invariance of the interval. Show that I 2 = Δt 2 – Δx 2 = Δt′ 2 – Δx′ 2 = I′ 2 (Hint: This is the hardest problem. Use values for x, t as in the diagram below; when using the Lorentz transformation, express things in a single frame; also, let c = 1 and set other velocities as fractions of c.)

Light cones A spacetime interval is not an ordinary distance. Intervals (unlike distances) can be positive, negative, or zero. Standard terminology: –If I 2 > 0, then the interval is time-like (eg, (O,e 1 )) –If I 2 = 0, then light-like (O,e 2 ) –If I 2 < 0, then space-like (O,e 3 )

Hyper-planes Recall the hyper-planes (ie, planes of simultaneity) from last week. The hyper-plane through the origin (event 1) is the set of all events that are simultaneous with event 1 in A’s frame of reference.

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