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 the primed observer, the velocity 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 tx to y at ty
One observer sees
Any other observer will 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 the translations 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 spacelike separated 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 timelike separated
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 Lorentz transformation is a rotation (around z):
In coordinates, this is
 is the angle of rotation
Another example of a valid Lorentz transformation is a boost (along x):
 is the rapidity

10. What a Boost Means
What is relationship between primed and unprimed observer for a boost?
Imagine an observer at the origin of space in the primed coordinates
Recalling that x0 = ct and thinking 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 in the x direction can be rewritten
From this you can get the familiar relationships

11. Sample Problem 11.1 (1)
Consider a boost along the x-axis followed by one along the y-direction by 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 boosts are given by
The combination of these 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-direction by 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 by trigonometric 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) tensor is 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 v and 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 lowering indices 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 and the 4D Levi-Civita tensor
We can make more invariant tensors 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 v and 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 antisymmetry of 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 proper time  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 a vector 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 with mass m1 and m2 collide elastically
Write down conservation of p0
Expand  using a Taylor series
The fourth component of momentum is energy! m1 m2 v1 v2
v'2
v'1

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 k k' p p' 

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? k k' p 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' 

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, for conventional 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 taking partial derivatives of 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, and to 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 which we 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 rest of 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 different components 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 in vacuum in a manifestly covariant manner
Ampere’s Law:
Raise the indices 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 these must 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 means when  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 expression is 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 means you 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 the last 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 boundary condition 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 can be 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

51. Sample Problem 11.9 (2) [formerly 6.1]
A point charge q moves along the z-axis at velocity v. Find the fields everywhere.
Recall that the primed coordinates are related to the unprimed coordinates by a Lorentz boost by v along the z-axis:
Note that ' = 