11. The Special Theory of Relativity

1. 11. The Special Theory of Relativity 11A. Time is the Fourth Dimension Symmetries of Space • We have already talked about some symmetries of space and time • Rotations and space translations are both good symmetries • These are all symmetries of Maxwell’s equations • Consider the distance formula • This distance is preserved by all these transformations • Laws of physics should be invariant under transformations that preserve distance

2. Symmetries With Time • There are some additional symmetries of conventional physics • Time translation,defined as • Pretty obviously, Maxwell’s equations in vacuum remain invariant • Galilean transformation: • Is this a symmetry of Maxwell’s equations in vacuum? • Suppose an object is following a trajectory x(t) • Then the velocity will be: • But according to theprimed observer, thevelocity will be: • However, according to Maxwell’s equations, light always moves at speed c • Maxwell’s equations are inconsistent with Galilean transformations

3. Constancy of Speed of Light • Conflict between Maxwell’s equations and Galilean transformation • In the presence of media, this is hardly surprising • Laws of physics look different if you are moving in a medium compared to standing still • It was attributed to “ether”, an undetected background medium • However, no experiment succeeded in finding this ether • Einstein speculated that the Galilean formula was wrong • He speculated that all observers would see the same speed of light • Let a light beam move from x at txto y at ty • One observer sees • Any other observerwill also see • So, whatever the correct formula, we must have (ty,y) (tx,x)

4. Time and the Distance Formula • Recall the conventional distance formula • Recall that rotations and space translations leave this invariant • But this formula doesn’t include time, and can’t explain why time translations nor Galilean translations work • In fact, Galilean transformations don’t work • Maybe we need to include time in this distance formula • The equation at top suggests a formula for the proper distance: • This distance is invariant under thetranslations we found before, and the rotations • But it is not invariant under Galilean transformations

5. Time is the Fourth Dimension • The appearance of time in this formula suggests time is another dimension • Actually, to make the units consistent, ct is the fourth dimension • We will put together time and three space dimensions into a vector we will collectively denote x, with components denoted by superscripts • The factor of c is a conversion constant to turn “seconds” into “meters” • No more significant than 12 inches/foot • The only difference between time and the other components is the minus sign in the proper distance formula • Points in spacetime are called “events” and have a location and a time

6. Proper Distance and Proper Time • Unlike the 3D distance formula, this quantity can have either sign • If s2 > 0, we say the events are spacelikeseparated and call s the proper distance • If an observer sees two events at the same time, s is the same as 3D distance • If s2 < 0, we say the points are timelikeseparated • We then define the proper time, given by • If an observer sees two events at the same place,  is the time difference between them • Either of these formulas can be thought of as the actual distance formula • We will use the proper time one • Known as the “mostly minus” convention

7. The Metric • Write this expression out explicitly in terms of all components: • This is long and awkward. • Define the metric tensor • We then summarize our formula as: • Einstein summation convention: • Any index that appears multiplied with index up and down is summed over • This simplifies our formula to:

8. Lorentz Transformations • What sort of coordinate transformations leave this expression invariant? • Easy to see that space or time translations will do so • How about four-dimensional “rotations”? • These are called homogenous Lorentz transformations • Or just “Lorentz transformations” for short • Not surprisingly, there are restrictions on  • Distance in primed frame will be • We want this to equal • To get these to match, we need: • Or, written as a matrix,

9. Lorentz Transformation Examples • One example of a valid Lorentztransformation is a rotation (around z): • In coordinates,this is •  is theangle of rotation • Another example of a valid Lorentztransformation is a boost (along x): • In coordinates,this is •  is the rapidity

10. What a Boost Means • What is relationship between primed andunprimed observer for a boost? • Imagine an observer at the origin ofspace in the primed coordinates • Recalling that x0 = ct andthinking of x1 as x, this tells us • So this is an observer moving in x-direction at • Often write things in terms of , defined by • Also useful to define • Then a boost inthe x direction can be rewritten • From this you can get the familiarrelationships

11. Sample Problem 11.1 (1) Consider a boost along the x-axis followed by one along the y-directionby an amount . Then do two more boosts in the same order, this time by an amount –. Show to second order in  that the combination is not the identity, but instead is a rotation. • The first two boostsare given by • The combination ofthese two is • The last two steps are the same thing with   –

12. Sample Problem 11.1 (2) Consider a boost along the x-axis followed by one along the y-directionby an amount . Then do two more boosts in the same order, this time by an amount –. Show to second order in  that the combination is not the identity, but instead is a rotation. • Multiply them • Compare to rotation around z: • Match to order  2 if  =  2

13. Infinitesimal Proper Time • If we are moving along with an object at uniform velocity, you see it not moving • Therefore  is the same as t for this frame •  is the time experienced by a moving object • What if an objects is moving non-uniformly? • For short periods of time, it is approximately linear, so we would have • Take the square root • Integrate over time to find proper time (ty,y) (tx,x)

14. Sample Problem 11.2 A particle accelerates uniformly along the x1 axis, so x1 = ½at2. Find a relationship between t and . How much proper time would it take to reach the speed of light? • We have • Substitute for the proper time: • The integral can be done bytrigonometric substitution, or by Maple • It reaches speed of light at at = c • This formula becomes nonsense for at > c • Though this looks like uniform acceleration, it is not uniform force • An object moving like this would experience rising acceleration

15. 11B. Vectors, Tensors, Covectors, Etc. Scalar, Vector, Tensor • Under a Lorentz transformation, coordinates transform as • A scalar is any quantity, that is unchanged by Lorentz transformation: • A (contravariant) vector is any quantity that transforms like x: • A (contravariant) (rank-2) tensoris any quantity that transforms as • We can generalize to make tensors of any rank • A scalar is a rank-0 tensor • A vector is a rank-1 tensor Sample Problem 11.3 Let vand w be any two vectors. Show the dot product is a scalar.

16. Covectors; Raising and Lowering Indices • Let us define a covector w as a linear mapping from vectors to scalars • Since it is linear, it must take the form • We can describe w by giving its components w • We can make a covector out of any vector. Define: • Then we have • Turning vector into a covector is called “lowering the index” • In components, we have • We can reverse this process to raise an index • First define the inverse metric g • Note: this is the same matrix as g • Then it is easy to see that • This process is called “raising the index” • This process of raising and loweringindices can be applied to any tensor

17. Sample Problem 11.4 What happens if we raise one or both indices of the metric tensor g? • Raise one index: • But gand g are inverse matrices • Raise two indices: • Confusing because g now has two meanings: • Let gin green mean metric with both indices raised • Let g in black mean inverse of g • So we have

18. Invariant Tensors and Covariant Equations • Certain tensors are invariant • When we perform a proper Lorentz transformation, they remain unchanged • These include the metric g andthe 4D Levi-Civita tensor • We can make more invarianttensors by raising or lowering indices,or by multiplying invariant tensors • An equation is covariant if, when true in one Lorentz frame, it automatically is true in all • An equation is manifestly covariant if it follows the following rules: • Any index that is repeated on any term is matched one up and one down • Any index that is not repeated on any term must occur on every term, on both sides of the equation • You can use invariant tensors whenever needed

19. Sample Problem 11.5 A rank two tensor T is proportional to the product of two vectors vand w. Write all possible expressions for T. • The tensor has two indices and the vectors have one each • We want some sort of expression like • We need to make this expression covariant • Some obvious terms • Some less obvious terms: • However, you can use antisymmetryof Levi-Civita to show

20. 11C. Momentum and Energy Time Dilation • Imagine an object following an arbitrary path through space-time • Proper time passes at a rate • If an object is moving at velocity v, then we have, by definition • We therefore have • Proper time always runs slower than coordinate time • But effects are tiny at low velocities

21. Four-Velocity • We want, so far as possible, to treat space and time on an equal footing • Consider, for example, the formula for velocity: • This formula is not wrong, because it’s a definition • This formula is horribly non-covariant • It has three components, not four • It treats x0 = ct as special, rather than just any other component • We want a concept like velocity, but applicable in 4D • We need to replace t with something that looks like t at low velocities • Recall, for slow velocities, t and propertime  run at almost the same rate • This suggests defining the four-velocity: • Writing it out explicitly, we have:

22. Four-Acceleration and Proper Acceleration • The four velocity is avector of constant length • We can similarly define the four-acceleration • Consider the Lorentz invariant quantity AA • For an observer temporarily moving along with an object, we would have • This is the acceleration experienced by the object itself • Called the “proper acceleration” • Since AA is Lorentz invariant, we can calculate this in any frame

23. Sample Problem 11.6 An object moves uniformly in a circle of radius R at angular frequency . What is the proper acceleration a? • Uniform circular motion looks like • The conventional speed is • And therefore • Rewrite all components of x in terms of : • Now work out the four-velocity and acceleration • So the proper acceleration is

24. Four-Momentum • Consider the conventional expression for momentum • This looks awful and non-covariant • Suppose a process conserves this momentum in one frame • There is no reason to expect it to conserve in another frame • It generally won’t • This suggests we need a different formula for momentum • A logical suggestion would be • It is now a vector • Suppose four-momentum is conserved in one frame • This is manifestly covariant; it will be true in all frames • In terms of conventional velocity, pis given by

25. Four-Momentum and Energy • All four components of momentum must be conserved • Only conserving three would not be covariant, and makes no sense • What is this fourth component of momentum? • Suppose we have two objects withmass m1 and m2 collide elastically • Write down conservation of p0 • Expand  using a Taylor series • The fourth component of momentum is energy! m1 v'1 v1 v2 m2 v'2

26. Comments on Four-Momentum • The momentum is of fixed magnitude • We can use this, for example, to show that Solving problems with conservation of 4-momentum • Name all the momenta in the problem • Write down conservation of 4-momentum • Rearrange cleverly • Dot the expression into itself • Simplify using pp = m2c2 • Write out any remaining momenta explicitly • Solve the remaining problem

27. Sample Problem 11.7 (1) A (massless) photon with energy E collides with an electron (mass m) and scatters at an angle  compared to the original direction. What is the final energy of the photon? • Call electron initial/final momentum p and p' • Call photon initial/final momentum k and k' • Conservation of momentum: • We don’t care about final electron, and don’t know much about it • Isolate p' • Dot this into itself • Use pp = m2c2 • So p' k p  k'

28. Sample Problem 11.7 (2) A (massless) photon with energy E collides with an electron (mass m) and scatters at an angle  compared to the original direction. What is the final energy of the photon? p' k p • Now write out momenta explicitly • Initial and final photon are massless • Call their energies E and E' • Let’s say initial photon is in the x1-direction • Final photon can be in the x1x2-plane • The initial electron is at rest, so • Write out all the dot-products • Substitute in • Solve for E'  k'

29. Newton’s Second and Third Laws • Everybody knows Newton’s second law • In conventional physics, this is equivalent to • But in relativity, p  mv • Therefore, one of these equations is false • Consider Newton’s third law • If we assume the momentum equation is correct, this implies • Which yields • And not the other • So we can get conservation of momentum if we assume • Of course, this still doesn’t look manifestly Lorentz invariant • Better would be something like • Or better still,

30. Confusion and Subtleties • Up to this point, we, and Jackson have been working in SI units • At this point Jackson switches to Gaussian units • This means this chapter’s equations are incompatible with previous chapters • We will, in contrast, continue in SI units • This makes some of our formulas slightly more complicated than Jackson’s • If you want to do it right, you should simply work in units where c = 1 • And while you’re at it, where 0 = 0 = 1 • An additional problem is that, for anything we’ve already defined in three dimensions, we have gotten used to writing with subscripts down • When we go to four dimensions, distinction between up and down • Whenever I write indices as 1, 2, 3, forconventional fields I’ll write them as index up • But when I use x, y, z, I’ll write them down • I will commonly use Latin indices (i, j, k) for indices that only take on values 1, 2, 3

31. 11D. Relativity with Electricity and Magnetism Fields and Partial Derivatives • We’ll be working with fields, which are functions of position • These can be scalar fields, vector fields, etc. • They transform like • We will also be takingpartial derivativesof functions • Certainly we know: • This equation is obviously covariant • Since it is presumably manifestly covariant, we realize partial derivative are like “down” indices • To help us remember this, andto save space, we will write:

32. Mixing Relativity with E and M • We expect the actual laws of E and M in vacuum to be relativistically invariant • Since we got some thing about relativity from E and M, we wouldn’t be surprised if they are already mostly correct • Many quantities we are currently using are not set up for relativity • We should have four-vectors, not three-vectors, for example Our goal: • Combine various quantities into scalars, vectors, tensors, etc. • Write equations of electricity and magnetism in manifestly invariant form • Change them as little as possible

33. The Four-Current • We have several vector quantities in E and M • J, E, B, A • We should only have four-vectors (or scalars, or tensors) • Often can make a reasonable guess about the fourth component by looking at suitable equations • Let’s start with J, for whichwe have conservation of charge • Write it out in components • We would expect this to somehow generalize to • It doesn’t take a genius to figure out that • So we have

34. The Four-Vector Potential • You might think the logical thing to try next would be E • Does it simply become a four-vector? • A point charge at rest produces an E, but no B • But a moving point charge produces both E and B • This suggests something more complicated is going on with E and B • Let’s try working on the vector potential A • This is tricky, because A can depend on choice of gauge • Some gauge choices, like Coulomb gauge, are not Lorentz invariant • Recall, it led to a scalar potential that instantaneously responded to charge • In Lorentz gauge, we found that fields react at speed of light • Lorentz gauge was defined by • This suggests

35. The Electromagnetic Tensor F • We have still to work out B and E • Let’s start by writing out components of E in terms of A: • Let’s define the electromagnetic tensor • It is trivial to see that • Then we note that • The diagonal components of F automatically vanish: • Therefore F has 43 = 12 non-zero components • 6 independent ones • Let’s work out the restof the non-zero ones: • We can summarize this as • i, j, k only go from 1 to 3

36. Putting it Together • We’re now ready to write down all of F: • First index is row, second is column • Recall, they are numbered 0, 1, 2, 3 • Fis a rank-2 anti-symmetric tensor • In relativity, B and E are differentcomponents of the same tensor • We can raise the first or second index of F • 1st index: rows 1 – 3 get minus • 2nd index: columns 1 – 3 get minus • And therefore

37. Maxwell’s Inhomogenous Equations • Let’s try to write Maxwell’s Equations invacuum in a manifestly covariant manner • Ampere’s Law: • Raise theindices on F • Easy to generalize: • What is the  = 0 equation? • Ampere’s Law and Coulomb’s Law collapse into a single equation

38. Gauss’s Law for Magnetism • Write out B = 0 • Two ways to write this in terms of F’s • The sum of thesemust also be zero • Write more succinctly using 3D Levi-Civita tensor • We then notice that • So we write this rule as • We therefore conjecture that the correct general formula is • If this is right, we still have to figure out what this meanswhen  is a space index

39. Faraday’s Law • What is this equation if  = i? • Because of anti-symmetry of Levi-Civita, one other index is zero • Other two we will call j and k, space indices • Use the identities • The last expressionis a curl, so this is • This is Faraday’s Law

40. Maxwell’s Equations Summarized • In relativity, Maxwell’s four equations become two • First one is Ampere’s law and Gauss’s law • Second one is Faraday’s law and no magnetic monopoles • Often, one works with a vector potential, in terms of which • In this case, second equation is automatic: • On second term, change indices     • Use anti-symmetry of  • Partial derivatives commute with each other, so

41. The Lorentz Force • One more fundamental equation of electromagnetism needs to be dealt with • We already argued that F should be dp/dt • Write this out in components • Multiply through by • A reasonable conjecture for the Lorentz force equation is • If you like F = ma, write this as

42. Sample Problem 11.8 (1) Find the general motion of a particle of mass m moving in a constant magnetic field , if it starts at the origin of space and time at  = 0 with velocity v. • We need to solve the equation • With both indices raised, we have: • But lower the second index, which meansyou get a minus sign on columns 1 – 3 • We now need to solve the equation • Which works out to • The initial four-velocity is

43. Sample Problem 11.8 (2) Find the general motion of a particle of mass m moving in a constant magnetic field , if it starts at the origin of space and time at  = 0 with velocity v. • The first two equations are trivial to solve • Define the cyclotron frequency • Substitute middle equation in thelast one • This has general solution • Substitute in middle: • Match boundary conditions at  = 0 • And substitute in:

44. Sample Problem 11.8 (3) Find the general motion of a particle of mass m moving in a constant magnetic field , if it starts at the origin of space and time at  = 0 with velocity v. • Recall that • So we just integrate these equations: • Subject to boundarycondition x(0) = 0 • If we want, we rewrite in terms of t

45. Gauge Transformations and Gauge Choice • A gauge transformation is any change of the form • Let’s try to write this in a covariant manner: • Gauges are not necessarily Lorentz invariant • Coulomb gauge, for example • Lorentz gauge, in contrast,is Lorentz invariant • In Lorentz gauge, we found • With a little work, these canbe combined to yield

46. Rewriting Solution in Lorentz Gauge (1) • This has integral over space, but not over time • To make it look more four-dimensional, add a time integral: • Now, recall for any -function • Applying this to t', we have • We therefore have: • Needed to restrict t' < t to make sure we get correct root in delta function

47. Rewriting Solution in Lorentz Gauge (2) • This is almost manifestly Lorentz covariant • Though this isn’t manifestly Lorentz covariant, it is Lorentz covariant • The delta function restricts x' to the light cone of the point x • Together with the Heaviside function, it refers to the past light cone of x

48. 11D. Lorentz Transformations in E and M It’s Not Complicated • We know how to transform anything • Suppose we have a four-current • Consider a boost in the x-direction • As viewed by the moving observer, we have • Write it all out: • In a general direction, this becomes • Don’t forget we must rotate coordinates too!

49. Lorentz Boost of EM-Fields • Consider an x-boost of the EM fields • Let’s work them out one at in massive parallel

50. Sample Problem 11.9 (1) [formerly 6.1] A point charge q moves along the z-axis at velocity v. Find the fields everywhere. • Let use move to a primed coordinate system,moving along with the particle at velocity v • For this observer, the fields are • Write E' in cylindrical coordinates • Now, perform a Lorentz boost by –v, back along the z-axis