Special RelativityChristopher R. PriorAccelerator Science and Technology CentreRutherford Appleton Laboratory, U.K.Fellow and Tutor in MathematicsTrinity College, Oxford

OverviewThe principle of special relativityLorentz transformation and consequencesSpace-time4-vectors: position, velocity, momentum, invariants, covariance.Derivation of E=mc2Examples of the use of 4-vectorsInter-relation between and , momentum and energyAn accelerator problem in relativityRelativistic particle dynamicsLagrangian and Hamiltonian FormulationRadiation from an Accelerating ChargePhotons and wave 4-vectorMotion faster than speed of light

ReadingW. Rindler: Introduction to Special Relativity (OUP 1991)D. Lawden: An Introduction to Tensor Calculus and RelativityN.M.J. Woodhouse: Special Relativity (Springer 2002)A.P. French: Special Relativity, MIT Introductory Physics Series (Nelson Thomes)Misner, Thorne and Wheeler: RelativityC. Prior: Special Relativity, CERN Accelerator School (Zeegse)

Historical backgroundGroundwork of Special Relativity laid by Lorentz in studies of electrodynamics, with crucial concepts contributed by Einstein to place the theory on a consistent footing.Maxwells equations (1863) attempted to explain electromagnetism and optics through wave theorylight propagates with speed c = 3108 m/s in ether but with different speeds in other framesthe ether exists solely for the transport of e/m wavesMaxwells equations not invariant under Galilean transformationsTo avoid setting e/m apart from classical mechanics, assumelight has speed c only in frames where source is at restthe ether has a small interaction with matter and is carried along with astronomical objects

Contradicted by:Aberration of star light (small shift in apparent positions of distant stars)Fizeaus 1859 experiments on velocity of light in liquidsMichelson-Morley 1907 experiment to detect motion of the earth through etherSuggestion: perhaps material objects contract in the direction of their motion

This was the last gasp of ether advocates and the germ of Special Relativity led by Lorentz, Minkowski and Einstein.

The Principle of Special RelativityA frame in which particles under no forces move with constant velocity is inertial.Consider relations between inertial frames where measuring apparatus (rulers, clocks) can be transferred from one to another: related frames.Assume:Behaviour of apparatus transferred from F to F' is independent of mode of transferApparatus transferred from F to F', then from F' to F'', agrees with apparatus transferred directly from F to F''.The Principle of Special Relativity states that all physical laws take equivalent forms in related inertial frames, so that we cannot distinguish between the frames.

SimultaneityTwo clocks A and B are synchronised if light rays emitted at the same time from A and B meet at the mid-point of AB

Frame F' moving with respect to F. Events simultaneous in F cannot be simultaneous in F'.Simultaneity is not absolute but frame dependent.

The Lorentz TransformationMust be linear to agree with standard Galilean transformation in low velocity limitPreserves wave fronts of pulses of light,

Solution is the Lorentz transformation from frame F (t,x,y,z) to frame F'(t',x',y',z') moving with velocity v along the x-axis:

Outline of Derivation

General 3D form of Lorentz Transformation:

Consequences: length contractionzMoving objects appear contracted in the direction of the motionRod AB of length L' fixed in F' at x'A, x'B. What is its length measured in F?Must measure positions of ends in F at the same time, so events in F are (t,xA) and (t,xB). From Lorentz:

Consequences: time dilationClock in frame F at point with coordinates (x,y,z) at different times tA and tB In frame F' moving with speed v, Lorentz transformation gives

So

Moving clocks appear to run slow

Schematic Representation of the Lorentz Transformation

v = 0.8cv = 0.9cv = 0.99cv = 0.9999c

Example: High Speed TrainObservers A and B at exit and entrance of tunnel say the train is moving, has contracted and has length

But the tunnel is moving relative to the driver and guard on the train and they say the train is 100 m in length but the tunnel has contracted to 50 m

All clocks synchronised. As clock and drivers clock read 0 as front of train emerges from tunnel.

Question 1As clock (and the driver's clock) reads zero as the driver exits tunnel. What does Bs clock read when the guard goes in?Moving train length 50m, so driver has still 50m to travel before he exits and his clock reads 0. A's clock and B's clock are synchronised. Hence the reading on B's clock is

Question 2What does the guards clock read as he goes in?

To the guard, tunnel is only 50m long, so driver is 50m past the exit as guard goes in. Hence clock reading is

Question 3Where is the guard when his clock reads 0?Guards clock reads 0 when drivers clock reads 0, which is as driver exits the tunnel. To guard and driver, tunnel is 50m, so guard is 50m from the entrance in the trains frame, or 100m in tunnel frame. So the guard is 100m from the entrance to the tunnel when his clock reads 0.

F(t,x) is frame of A and B, F'(t',x') is frame of driver and guard.

Question 1As clock reads zero as the driver exits tunnel. What does Bs clock read when the guard goes in?Repeat within framework of Lorentz transformation

Question 2What does the guards clock read as he goes in?F(t,x) is frame of A and B, F'(t',x') is frame of driver and guard.

Question 3Where is the guard when his clock reads 0?F(t,x) is frame of A and B, F'(t',x') is frame of driver and guard.Or 100m from the entrance to the tunnel

Question 4Where was the driver when his clock reads the same as the guards when he enters the tunnel?F(t,x) is frame of A and B, F'(t',x') is frame of driver and guard.Or 100m beyond the exit to the tunnel

Example: Cosmic Raysm-mesons are created in the upper atmosphere, 90km from earth. Their half life is =2 s, so they can travel at most 2 10-6c=600m before decaying. So how do more than 50% reach the earths surface?Mesons see distance contracted by , so

Earthlings say mesons clocks run slow so their half-life is and

Both give

Space-timeAn invariant is a quantity that has the same value in all inertial frames.Lorentz transformation is based on invariance of

4D space with coordinates (t,x,y,z) is called space-time and the point is called an event.Fundamental invariant (preservation of speed of light):

txAbsolute futureAbsolute pastConditional present

4-VectorsThe Lorentz transformation can be written in matrix form asAn object made up of 4 elements which transforms like X is called a 4-vector(analogous to the 3-vector of classical mechanics)

InvariantsBasic invariant Inner product of two 4-vectors Invariance:

4-Vectors in S.R. MechanicsVelocity:

Note invariant

Momentum

Example of Transformation: Addition of VelocitiesA particle moves with velocity in frame F, so has 4-velocityAdd velocity by transforming to frame F to get new velocity corresponding to 4-vector Lorentz transformation gives

4-ForceFrom Newtons 2nd Law expect 4-Force given by

Einsteins Relation Momentum invariantDifferentiateE=mc2 is total energy

Basic Quantities used in Accelerator Calculations

Velocity v. Energy

Energy-Momentum InvariantExample: ISIS 800MeV protons (E0=938MeV)=> pc=1.463GeV

Relationships between small variations in parameters E, T, p, , (exercise)Note: valid to first order only

4-Momentum ConservationEquivalent expression for 4-momentum

Invariant

Classical momentum conservation laws conservation of 4-momentum. Total 3-momentum and total energy are conserved.

A body of mass M disintegrates while at rest into two parts of rest masses M1 and M2. Show that the energies of the parts are given by

SolutionBefore:After:Conservation of 4-momentum:

Example of use of invariantsTwo particles have equal rest mass m0.

Frame 1: one particle at rest, total energy is E1.

Frame 2: centre of mass frame where velocities are equal and opposite, total energy is E2.

Problem:Relate E1 to E2

Total energy E1(Fixed target experiment)Total energy E2(Colliding beams expt)

Collider ProblemIn an accelerator, a proton p1 with rest mass m0 collides with an anti-proton p2 (with the same rest mass), producing two particles W1 and W2 with equal rest mass M0=100m0Expt 1: p1 and p2 have equal and opposite velocities in the lab frame. Find the minimum energy of p2 in order for W1 and W2 to be produced.Expt 2: in the rest frame of p1, find the minimum energy E' of p2 in order for W1 and W2 to be produced.

Note: same m0, same p mean same E.Total 3-momentum is zero before collision and so is zero afterwards.Energy conservation E=E > rest energy = M0c2 = 100 m0c2

Use previous result to relate E1 to total energy E2 in C.O.M frame

4-Acceleration

Radiation from an accelerating charged particleRate of radiation, R, known to be invariant and proportional to in instantaneous rest frame.But in instantaneous rest-frameDeduce

Rearranged: Relativistic Larmor Formula

Motion under constant acceleration; world linesIntroduce rapidity r defined by

ThenAndSo constant acceleration satisfies

Particle Paths

World line of particle is hyperbolicNon-relativistic approximation

Relativistic Lagrangian and Hamiltonian Formulation3-force eqn of motion under potential V:Standard Lagrangian formalism:Since , deduceRelativistic Lagrangian

Hamiltons equations of motionSince

Photons and Wave 4-VectorsMonochromatic plane wave:

Phase is the number of wave crests passing an observer, an invariant.

Position 4-vector, XWave 4-vector, K

Relativistic Doppler ShiftFor light rays, phase velocity is

So where is a unit vectorLorentz transformNote: transverse Doppler effect even when

Motion faster than lightTwo rods sliding over each other. Speed of intersection point is v/sin, which can be made greater than c.Explosion of planetary nebula. Observer sees bright spot spreading out. Light from P arrives t=d2/2c later.

Light was assumed to propagate through the ether, which permeated all space, was of negligible density and had negligible interaction with matter. The ether existed solely for the transport of electromagnetic waves, which set e/m apart from the rest of physics. It was accepted that the laws of mechanics were invariant under Galilean transformations.The existence of the ether meant that the laws of e/m were not invariant. There existed a preferred system in which the velocity of light was c; in other systems it was not equal to c.

To avoid setting e/m apart from the rest of physics, it was postulatedThat v=c in systems where the source is at restThe preferred reference frame is the system in which the medium of propagation is at restThe ether does have a small interaction with matter and can be carried along by astronomical objects such as the earth.

The downfall came from experiments.Aberration of starlight: the small shift in position of distant stars is readily explained by the motion of the earth round the sun, and contradicts the hypothesis that the velocity of light is determined by the transmitting medium (our atmosphere) or that the ether is dragged along. In neither case would aberrations occur.Fizeaus experiments on the velocity of light in liquids flowing in a pipe either in the direction of or against the direction of propagation. Results can only be explained if small bodies also drag the ether along with them. Newton: 1. a rigid body has the same size in all frames, and 2. time is absolute.Einstein: 1. the velocity of light is finite, and 2. the velocity of light has the same value in all frames.Lorentz lived from 1853 to 1928 and became known as the "Grand Old Man of Dutch Physics". He came up with the idea that the laws of nature must be invariant to a change of coordinate systems. The consequences of this were that time and space variables needed to enter into equations on an equal footing (same order of differentiation, etc.). Maxwell's Equations were readily made to be "Lorentz-invariant" and were then seen to be somehow very fundamental. These ideas laid the basis for the theory of special relativity a few years later. But new theories were always tested against this condition - if it wasn't Lorentz-invariant, something must be badly wrong. Schrdinger ran into this problem when he first tackled quantum mechanics.Hendrik Antoon Lorentz was born at Arnhem, The Netherlands, on July 18, 1853, as the son of nursery-owner Gerrit Frederik Lorentz and his wife ne Geertruida van Ginkel. Spent his life as Professor of Physics at Leyden University.

From the start of his scientific work, Lorentz took it as his task to extend James Clerk Maxwell's theory of electricity and of light. Already in his doctor's thesis, he treated the reflection and refraction phenomena of light from this standpoint which was then quite new. His fundamental work in the fields of optics and electricity has revolutionized contemporary conceptions of the nature of matter. In 1878, he published an essay on the relation between the velocity of light in a medium and the density and composition thereof. The resulting formula, proposed almost simultaneously by the Danish physicist Lorenz, has become known as the Lorenz-Lorentz formula. Lorentz also made fundamental contributions to the study of the phenomena of moving bodies. In an extensive treatise on the aberration of light and the problems arising in connection with it, he followed A. J. Fresnel's hypothesis of the existence of an immovable ether, which freely penetrates all bodies. This assumption formed the basis of a general theory of the electrical and optical phenomena of moving bodies. From Lorentz stems the conception of the electron; his view that his minute, electrically charged particle plays a rle during electromagnetic phenomena in ponderable matter made it possible to apply the molecular theory to the theory of electricity, and to explain the behaviour of light waves passing through moving, transparent bodies. The so-called Lorentz transformation (1904) was based on the fact that electromagnetic forces between charges are subject to slight alterations due to their motion, resulting in a minute contraction in the size of moving bodies. It not only adequately explains the apparent absence of the relative motion of the Earth with respect to the ether, as indicated by the experiments of Michelson and Morley, but also paved the way for Einstein's special theory of relativity. It may well be said that Lorentz was regarded by all theoretical physicists as the world's leading spirit, who completed what was left unfinished by his predecessors and prepared the ground for the fruitful reception of the new ideas based on the quantum theory. In 1919, he was appointed Chairman of the Committee whose task it was to study the movements of sea water which could be expected during and after the reclamation of the Zuyderzee in The Netherlands, one of the greatest works of all times in hydraulic engineering. His theoretical calculations, the result of eight years of pioneering work, have been confirmed in actual practice in the most striking manner, and have ever since been of permanent value to the science of hydraulics.