Special Relativity and Field Theory

4/14/2008

Stanford class with Leonard Susskind. Lecture 1.

Inertial reference frames

Inertial reference frames are:

not accelerated

F=ma is valid

laws of physics are the same in any inertial reference frame

However, Maxwell’s equations did not predict constant velocity of light. For details look online at Stanford iTunes from about a year ago:

iTunes U/Stanford/Science and Technology/Modern Theoretical Physics – Fall 2006

Brief review of Special Relativity

Consider two inertial frames moving with respect to each other in the (x,t) plane. Coordinates in one frame transform to the other via a Galilean rotation.

Use to have uniform units for all coordinates. Makes sense in the same way that x,y,z coordinates should use the same units.

A Galilean transformation assumes that time is universal and has the same value in all inertial systems. Consider the path of light in the (x,t) plane,

According to the transformation equations, equation ,

then

This transformation rule contradicts the Michelson-Morley experiment which demonstrated that the velocity of light is constant.

Circular Trig

Consider a rotation in the (x,y) plane.

where

so that

then

and

Notice that the distance is invariant under the transformation above

then

The Galilean transformation works for small velocities. According to Newton, the speed of light is large, but a constant in every inertial reference frame, counter to the Michelson-Morley experiment. Recall that in the year of the experiment 1887 no one had traveled 100 miles an hour and lived to tell the tale.

Hyperbolic Trig

Let . The coordinates for light allow two possibilities (directions),

Both directions are allowed as roots to the quadratic equation

This equation is necessary and sufficient for the constancy of the speed of light. This equation applies only for the speed of light.

The problem is now to find a transformation that leaves the velocity of light invariant.

Generalize the problem to any velocity, not just the velocity of light, so drop the condition of zero value and state the invariance.

Invariance requires the use of hyperbolic functions rather than circular functions. Replace with .

Define

then

Compare the ranges of and .

Guess the following transformation (Lorentz transformation)

Verify the expected invariant.

The “angle” parameter depends upon the velocity between the two frames. in both frames so that

Maxwell was aware that his equation had a rest frame where light went the same speed left and right. Maxwell assumed that we were approximately at rest with respect to the Ether. He assumed that the Ether was necessary.

The Michelson-Morley experiment showed constancy for the velocity of light in Earth’s orbit in any direction.

Find the interpretation of the parameter .

after transformation

therefore

Invert to find in terms of v. Start the inversion procedure by first squaring both sides of the equation.

From above

Therefore

Now eliminate for v.

Restore the velocity of light to the equations.

Exercise: calculate compounding velocities. Hint, add the hyperbolic angles, not the velocities.

is the distance along the hyperbola.

The Lorentz transformation reduces to Newton’s velocity addition for small velocities.

Investigate time transformation.

therefore, no agreement as to simultaneity.

Substitute

to solve the inverse problem.

Consider the differential distance.

is independent of (invariant under) a rotation of the axis.

Consider the hyperbolic distance element. is the proper time.

is invariant under a Lorentz transformation and is independent of .

The proper time elapsed does not depend upon the choice of coordinates and so is an invariant under a Lorentz transformation. Proper time is the time an observer measures with a clock traveling with the observer.

The twin paradox: see other lectures.

The shortest path with the hyperbolic distance element is not the same as a straight line with a circular distance element.