Notes on Einstein

Chapter 1

• It makes no sense to ask if the axioms of geometry are true.
• If we add a bridge claim, e.g., that two points on a practically rigid body correspond to a line segment, geometry turns into physics.

Chapter 2

• Position in terms of place on rigid body
• Coordinate body: counterfactual measurements of length

Chapter 3

• "Space" is vague. Instead, measure positions against a rigid body. (Why are bodies not just as vague as space? What does it mean that they are rigid? -- Circularity?)
• We want to graph paths of objects in space and time. Do so by having a clock attached to the rigid body. Problem: Light from object propagates at finite speed.

Chapter 4

• Stars move in circles relative to earth. Law of inertia only holds for "Galilean" or "inertial" coordinates.
• 'A system of co-ordinates of which the state of motion is such that the law of inertia holds relative to it is called a " Galileian system of co-ordinates." The laws of the mechanics of Galflei-Newton can be regarded as valid only for a Galileian system of co-ordinates.' -- Note the circularity. Is it virtuous?

Chapter 5

• If a coordinate system K' is moving (i.e., defined by a moving body) uniformly and in a straight line relative to K, then something moves uniformly and in a straight line with respect to one system iff it does so with respect to the other. Hence, if one is inertial, so is the other.
• The mechanical laws of Newton hold equally in both systems: Principle of Relativity (in the restricted sense).
• Just for Newtonian laws? "But that a principle of such broad generality should hold with such exactness in one domain of phenomena, and yet should be invalid for another, is a priori not very probable."
• If the principle fails, then there is a system in which the laws are particularly simple, and that system we'd call "absolutely at rest".
• We don't notice differences in phenomena based on how we are oriented with respect to the earth's 30km/s motion. (Maybe this motion isn't big enough?)

Chapter 6

• Addition of velocities. Seems self-evident that in a minute the train moves 500 m, and the man walks 50 m on the train, then the man moves 550 m relative to the embankment, so that W = v + w.
• But this is false.

Chapter 7

• Light propagates at about 300,000 km/s.
• This does not depend on the motion of the emitter!
• But now we have a problem. Send out beam along embankment. Travel at, say, 100,000 km/s (very fast train). Should see light traveling at 200,000 km/s relative to train.
• It seems we must abandon relativity or the law of propagation of light. "Those of you who have carefully followed the preceding discussion are almost sure to expect that we should retain the principle of relativity, which appeals so convincingly to the intellect because it is so natural and simple."
• But there is a reconciliation.

Chapter 8

• Light strikes rails at A and B. Simultaneously. What does that mean?
• Self evident!
• But that won't do--we must have an empirical test. "The concept does not exist for the physicist until he has the possibility of discovering whether or not it is fulfilled in an actual case. ... As long as this requirement is not satisfied, I allow myself to be deceived as a physicist (and of course the same applies if I am not a physicist), when I imagine that I am able to attach a meaning to the statement of simultaneity. (I would ask the reader not to proceed farther until he is fully convinced on this point.)"
• Can we charitably read this as a metascientific constraint on scientific theorizing?
• One option (not considered by text): Set two clocks in one location, carry one to the other location, and see if events happen at the same time.
  • Why not? Well, one problem is that we want simultaneity to be symmetric. But what if we get different results depending on which place we start? If we have empirical data that it doesn't matter, that's OK. But the fact that we need to advert to empirical data shows... Well, what does it show? That this doesn't capture our concept of simultaneity, where symmetry is trivial? But maybe part of the framework only within which the concept works is that the empirical data come out right.
• Instead, we stipulatively define simultaneity in terms of mirrors and flashes. If it's stipulative, why is it better than the previous definition? (No need for clocks. Symmetry. Fits better with principle of relativity--no special frame.)
• Physical hypothesis: Identical clocks run at the same rate: if initially simultaneous, always simultaneous. (Does this assume determinism? Or just that there are some such clocks?)
• Physical hypothesis: transitivity.

Chapter 9

• Get different results relative to embankement and relative to train.

Chapter 10

• How to measure the length of the train from the embankment?
• Put clocks along embankment and mark off where the two points on the train correspond to, and then measure at will.
• A priori uncertain we'll get the same measurement

Chapter 11

• The conflict about relativity and uniform propagation of light now disappears, because two assumptions have been undercut.
• Suppose that the primed coordinate system is moving with velocity v along x axis. Then, on Galilean-Newtonian assumptions, x' = x-vt, y'=y, z'-z. But:
• x' = (x-vt)/(1-(v/c)²)^1/2, t'=(t-vx/c²)/(1-(v/c)²)^1/2.
• Light correctly goes at c. Try: x=ct. Then x'=ct'.

Chapter 12

• Let t=0. Take a rod of length 1 in moving frame: x'=0 to x'=1. Then, x ranges from 0 to (1-(v/c)²)^1/2. In other words, it's shorter. E.g., if v=c/2, then the length in the embankement frame is 0.86. If v=100km/h, then the length is about 1-4x10^-15.
• Consider clock on train at x=vt (i.e., moving with train). Ticks at t'=0 and t'=1. I.e., t=0 and t=1/(1-(v/c)²)^1/2. So the clock moves faster. If v=c/2, the tick length is 1.163.

Chapter 13

• Addition of velocities. In Galilean system, easy: x'=wt', so x=(v+w)t, and velocity is v+w.
• Let x'=wt'. Get x=Wt, W=(v+w)/(1+vw/c²). If w is c, then W=(c+v)/(1+v/c)=c. If w is -c, then W=(-c+v)/(1-v/c)=-c.

Chapter 14

• Laws are covariant with respect to Lorentz transformations.

Chapter 15

• Kinetic energy is mc²/((1-(v/c)²)^1/2. We'll need more and more energy to get closer to velocity c.

Chapter 16

• Empirical data.
• Can also be explained by positing contractions of objects.

Chapter 17

(Need to talk about cones and reference frames.)

Chapter 18

• General principle of relativity
• Problem: Braking in a train

Chapter 19

• Weight and mass are related

Chapter 20

• Chest experiment: gravitational field
• The braking in a train is equivalent to a gravitational field

Chapter 21

• Whenever there is a difference in behavior, there must be a relevant difference in the situation explaining it (hence, determinism). Why do different reference bodies yield different behavior?

Chapter 22

• Light-rays don't move in straight lines vis-a-vis accelerated observers. Hence neither do they move in straight lines through gravitational fields.

Chapter 23

• Clock near outside of spinning disc goes more slowly. The rod near the outside of the spinning disc will be shorter. So, the circumference will be more than 2πr. (Likewise, the Pythagorean theorem will not hold.)

Chapter 28

• The Principle of Relativity induces a "comprehensive limitation on the laws of nature".

Appendix I

• Assumptions: speed of light same in all frames
• K looks from K' just as K' from K
• Isotropy of space and time