Relativity Chapter 1

RelativityChapter 1

A Brief Overview of Modern Physics 20th Century revolution: - 1900 Max Planck Basic ideas leading to Quantum theory - 1905 Einstein Special Theory of Relativity 21st Century Story is still incomplete

Basic Problems • Newtonian mechanics fails to describe properly the motion of objects whose speeds approach that of light • Newtonian mechanics is a limited theory • It places no upper limit on speed • It is contrary to modern experimental results • Newtonian mechanics becomes a specialized case of Einstein’s special theory of relativity when speeds are much less than the speed of light

Galilean Relativity • To describe a physical event, a frame of reference must be established • There is no absolute inertial frame of reference • This means that the results of an experiment performed in a vehicle moving with uniform velocity will be identical to the results of the same experiment performed in a stationary vehicle

Galilean Relativity • Reminders about inertial frames • Objects subjected to no forces will experience no acceleration • Any system moving at constant velocity with respect to an inertial frame must also be in an inertial frame • According to the principle of Galilean relativity, the laws of mechanics are the same in all inertial frames of reference

Galilean Relativity • The observer in the truck throws a ball straight up • It appears to move in a vertical path • The law of gravity and equations of motion under uniform acceleration are obeyed

Galilean Relativity • There is a stationary observer on the ground • Views the path of the ball thrown to be a parabola • The ball has a velocity to the right equal to the velocity of the truck

Galilean Relativity – conclusion • The two observers disagree on the shape of the ball’s path • Both agree that the motion obeys the law of gravity and Newton’s laws of motion • Both agree on how long the ball was in the air Conclusion: There is no preferred frame of reference for describing the laws of mechanics

Frames of Reference and Newton's Laws The cornerstone of the theory of special relativity is the Principle of Relativity:The Laws of Physics are the same in all inertial frames of reference.We shall see that many surprising consequences follow from this innocuous looking statement.

Let us review Newton's mechanics in terms of frames of reference. A point in space is specified by its three coordinates (x,y,z) and an "event" like, say, a little explosion by a place and time – (x,y,z,t). A "frame of reference" is just a set of coordinates - something you use to measure the things that matter in Newtonian mechanical problems - like positions and velocities, so we also need a clock.

The "laws of physics" we shall consider are those of Newtonian mechanics, as expressed by Newton's laws of motion, with gravitational forces and also contact forces from objects pushing against each other._____________________________ For example, knowing the universal gravitational constant from experiment (and the masses involved), it is possible from Newton's second law, force = mass x acceleration, to predict future planetary motions with great accuracy.

Suppose we know from experiment that these laws of mechanics are true in one frame of reference. How do they look in another frame, moving with respect to the first frame? To figure out, we have to find how to get from position, velocity and acceleration in one frame to the corresponding quantities in the second frame. Obviously, the two frames must have a constant relative velocity, otherwise the law of inertia won't hold in both of them.

Let's choose the coordinates so that this velocity is along the x-axis of both of them.

Notice we also throw in a clock with each frame. Now what are the coordinates of the event (x,y,z,t) in S'? It's easy to seet' = t - we synchronized the clocks when O‘ passed O. Also, evidently, y' = y and z' = z, from the figure. We can also see that x = x' +vt. Thus (x,y,z,t) in S corresponds to (x',y',z', t') in S', where That's how positions transform - these are known as the Galilean transformations.

What about velocities ? The velocity in S' in the x' direction This is just the addition of velocities formula

How does acceleration transform?

the acceleration is the same in both frames. This again is obvious - the acceleration is the rate of change of velocity, and the velocities of the same particle measured in the two frames differ by a constant factor - the relative velocity of the two frames. Since vis constant we have

If we now look at the motion under gravitational forces, for example, we get the same law on going to another inertial frame because every term in the above equation stays the same. Note that acceleration is the rate of change of momentum - this is the same in both frames. So, in a collision, if total momentum is conserved in one frame (the sum of individual rates of change of momentum is zero) the same is true in allinertial frames.

Maxwell’s Equations of Electromagnetismin Vacuum Gauss’ Law for Electrostatics Gauss’ Law for Magnetism Faraday’s Law of Induction Ampere’s Law

The Equations of Electromagnetism ..monopole.. Gauss’s Laws 1 ? 2 ...there’s no magnetic monopole....!!

The Equations of Electromagnetism .. if you change a magnetic field you induce an electric field......... Faraday’s Law 3 Ampere’s Law 4 .. if you change an electric field you induce a magnetic field.........

Faraday’s law:dB/dt electric field Maxwell’s modification of Ampere’s law dE/dt magnetic field ˆ j ˆ z E(x, t) = EP sin (kx-t) B(x, t) = BP sin (kx-t) Electromagnetic Waves These two equations can be solved simultaneously. The result is:

E(x, t) = EP sin (kx-t) B(x, t) = BP sin (kx-t) ˆ j ˆ z Plane Electromagnetic Waves Ey Bz c x

E(x, t) = EP sin (kx-t) B(x, t) = BP sin (kx-t) ˆ j ˆ z Plane Electromagnetic Waves Ey Bz Notes: Waves are in Phase, but fields oriented at 900. k=2π/λ. Speed of wave is c=ω/k (= fλ) c x At all timesE=cB

It was recognized that the Maxwell equations did not obey the principles of Newtonian relativity. i.e. the equations were not invariant when transformed between the inertial reference frames using the Galilean transformation. Lets consider an example of infinitely long wire with a uniform negative charge density λ per unit length and a point charge qlocated a distance y1 above the wire.

The observer inSandS’see identical electric field at distancey1=y1’from an infinity long wire carrying uniform chargeλper unit length. Observers in bothSandS’measure a force on chargeqdue to the line of charge.

However, the S’ observer measured and additional force due to the magnetic field at y1’ arising from the motion of the wire in the -x’ direction. Thus, the electromagnetic force does not have the same form in different inertial systems, implying that Maxwell’s equations are not invariant under a Galilean transformation.

Speed of the Light It was postulated in the nineteenth century that electromagnetic waves, like other waves, propagated in a suitable material media, called the ether. In according with this postulate the ether filed the entire universe including the interior of the matter. It had the inconsistent properties of being extremely rigid (in order to support the stress of the high electromagnetic wave speed), while offering no observable resistance to motion of the planet, which was fully accounted for by Newton’s law of gravitation.

Speed of the Light The implication of this postulate is that a light wave, moving with velocity c with respect to the ether, would travel at velocity c’=c +v with respect to a frame of reference moving through the ether at v. This would require that Maxwell’s equations have a different form in the moving frame so as to predict the speed of light to be c’, instead of

Conflict Between Mechanics and E&M A. Mechanics Galilean relativity states that it is impossible for an observer to experimentally distinguish between uniform motion in a straight line and absolute rest. Thus, all states of uniform motion are equal.

Conflict Between Mechanics and E&M B. E&M Initially- The initial interpretation of the speed of light in Maxwell's theory was this cwas the speed of light seen by observers in absolute rest with respect to the ether. In other reference frames, the speed of light would be different from c and could be obtained by the Galilean transformation.

Problem- It would now be possible for an observer to distinguish between different states of uniform motion by measuring the speed of light or doing other electricity, magnetism, and optics experiments.

Possible Solutions 1. Maxwell's theory of electricity and magnetism was flawed. It was approximately 20 years old while Newton's mechanics was approximately 200 years old. 2. Galilean relativity was incorrect. You can detect absolute motion! 3. Something else was wrong with mechanics (I.e Galilean transformation).

Experimental Results Most physicists felt that Maxwell's equations were probably in error. Numerous experiments were performed to detect the motion of the earth through the ether wind. The most famous of these experiments was theMichelson-Morleyexperiment. Because of the tremendous precision of their interferometer, it was impossible for Michelsonand Morley to miss detecting the effect of the earth's motion through the ether unless mechanics was flawed!

The Michelson-Morley experiment is a race between light beams. The incoming light beam is split into two beams by a half-silvered mirror. The beams follow perpendicular paths reflecting off full mirrors before recombining back at the half mirror. Time differences are seen in the interference pattern on the screen.

Theory We will simplify the calculations by assuming that L1 = L2 = L. The time required to complete path 1 (horizontal path) is given by where we have used the Galilean transformation and velocity = distance/time.

Since u<<c, we can use the binomial approximation:

We can determine the time required to complete path 2 (vertical path) using the distance diagram below: u(T2/2) u(T2/2) Using the Pythagorean theorem, we have:

Again, using the binomial approximation:

Thus, the time difference for the two paths is approximately We can now calculate the phase shift in terms of wavelengths as follows:

Thus, the phase shift in terms of a fraction of a wavelength is given by

Using a sodium light,  = 590 nm, and a interferometer with L = 11 m, we have This was a very large shift (20%) and couldn't have been over looked.

Result - No shift was ever observed regardless of when the experiment was performed or how the interferometer was orientated!

Einstein’s Postulates • In 1905 Albert Einstein published a paper on the electrodynamics of moving bodies. In this paper, he postulated that absolute motion can not be detected by any experiment. That is, there is no ether. The reference frame connected with earth is considered to be at rest and the velocity of the light will be the same in any direction. His theory of special relativity can be derived from two postulates: • Postulate 1:Absolute uniform motion can not be detected. • Postulate 2:The speed of light is independent of the motion of the source.

The Lorentz Transformation Galilean transformations: x = x’ + vt’, y = y’, z = z’, t = t’ The inverse transformations are x’ = x – vt, y’ = y, z’ = z, t’ = t These equations are consistent with experimental observations as long as v is much less than c. They lead to the familiar classical addition law for velocities. If a particle has velocity ux = dx/dt in frame S, its velocity in frame S’ is from here we have

The Lorentz Transformation It should be clear that the Galilean transformation is not consistent with Einstein’s postulates of special relativity. If light moved along thex axis with speed ux’=cinS’, these equations imply that the speed in S is ux=c+v rather than ux=c, which is not consistent with Einstein’s postulates and experiment. The classical transformation equations must therefore be modified .

The Lorentz Transformation We assume that the relativistic transformation equation for x is the same as the classical equation except for a constant multiplier on the right side: whereγ is a constant that can depend on v and c but not on coordinates. The inverse transformation in this case

Lorentz Transformation The transformation described by these equations is called the Lorentztransformation. Lorentz’s equations replace the flawed Galileo transformation equations in relating the measurements of two different observers in uniform motion relative to each other.

Relativity Chapter 1