• E= mc²
• World lines
• 4-dimensional physics
• What does "mass" mean?
• Causality

The unification of electricity and magnetism

Einstein’s one simple postulate solves a lot of problems. Consider the magnetic force on a moving charge due to the electric current in an electrically neutral wire (no electric field):

The magnetic force occurs when the charge is moving. If we look at it from the charge’s point of view (i.e., in its own "rest frame"), there can’t be a magnetic force, but there must be some kind of force, because the charge is accelerating.

So, the principle of relativity tells us that the charge must see an electric field in its rest frame. (Why must it be an electric field?) How can that be? The answer comes from Lorentz contraction. The distances between the + and - charges in the wire are Lorentz contracted by different amounts because they have different velocities. The wire appears to have an electrical charge density. (The net charge in a current loop will still be zero, but the opposite charge is found on the distant part of the loop, where the current flows the opposite direction.)

So net forward momentum is lost to the light. The lost momentum is: (v/c) DE/c, where DE is the lost energy.

v(Dm) +(Dv) m = (v/c²) DE.

So we can figure out a relation between the lost mass and the lost energy, if only somehow we can figure out how much the star's velocity is dropping, i.e. what's Dv?

The one thing you know easily is Dv = 0, because our choosing some reference frame with v ¹ 0 can't make the star accelerate.

There is no preferred reference frame: we mean it and we use it all the time in calculations.

So the Dv drops out of the equation

v(Dm) +(Dv) m = (v/c²) DE.

giving: Dm = DE/c²

m = E/c²

This does NOT say that "mass is convertible to energy". It says that mass and energy are two different words for the same thing, measured in units that differ by a factor of c².

A word on this equation. The mass appearing in this equation is the same mass that appears in the equation p=mv. We take it as a definition that m is the constant of proportionality between the momentum and the velocity. If this is true, then m=m_0*time dilation factor (1/sqrt(1-v^2/c^2)). m_0 is called the rest mass and m the relativistic mass. If we accept this definition, then we have to give up on F=ma. It turns out the mass in this equation will depend on the direction. That is, mass must be thought of as a vector if we use Newton's second law. There are two main conventions in physics: 1) m=m_0*time dilation factor and 2) m is an invariant given by m^2=(E^2-(pc)^2)/c^2. The high energy physicists use the latter and everybody uses the former. It does not matter which one you use. Just be consistent.

So on the first account mass is not an invariant. Is there anything "real" about an object, i.e. not dependent on how you look at it?

Old invariant: m

New invariant: E²-p²c²= m₀²c⁴

Then how can they have momentum? We cannot really understand this until we introduce quantum mechanics. However, let's imagine an object which has E/c = p. Then, (E/c)² - p² = 0. Does this mean that this object has no invariant mass? Is this possible?

• The "speed of light" is not a property unique to light. Light moves with this magic speed because it has no rest mass.
• Newtonian physics does not allow massless objects. They would always have zero energy and momentum, and would therefore be unobservable. SR allows such objects to exist, but they must always move at speed c. Light is not the only such object. Neutrinos were recently thought to be this way also. (If they do have any mass, it is very small.)
• A massless object always moves at speed c. It is easily verified that all colors of light (and radio pulses, etc.) from distant objects (eg, quasars) get to us after the same transit time. (But now some versions of string theory claim that very high-frequency light might show some slight frequency-dependent speed, in a testable range!)

I can measure the energy and momentum of the stuff inside by letting the box collide with other objects (assume the box itself to be very light so we can ignore its energy and momentum). Suppose that when the box is at rest (p=0), I measure energy E_o. Therefore, the rest mass of the stuff is given by
(m₀c)² = (E_o/c)² - 0². That is, E_o = m₀c².

Now ask a related question:

The reason why N’s concept works so well is, as usual, that unless we are dealing with objects moving at or near the speed of light, the effects are tiny. For example, in a macroscopic object at room temperature, the extra mass due to the thermal energy (in SR, all energy contributes to mass) is about 10^-10 of the mass of the atoms. That’s why Newton was able to assume conservation of mass. In the Newtonian limit, mass does tell us "the quantity of matter" (i.e., the total number of protons and neutrons), but this is only an approximation.

In 1908, Minkowski realized that there is a beautiful geometrical interpretation of SR. It is based on the mathematical similarity between rotations and Lorentz transformations. However, you need not stick to this presentation. By far the simplest presentation is that of Rohrlich's on pages 76-86. In fact, this is what I will present in class.

• Space and time are unified (but, beware of the - sign: the time axis is different from the space axes). This is partial recompense for the loss of absolute space and time.
• A lot of our 3-d thinking, especially about vectors, is useful in thinking about things in 4-d. More on this next time.
• The universal speed, c, now has a geometrical meaning. It is merely the conversion factor between space and time units. To see this, go back to 3-d. Suppose that we measured x in meters and y in feet:

The geometrical interpretation of SR helped lead Einstein to general relativity. This is a prime example of the value of theories beyond explaining old phenomena and predicting new ones. Mathematically equivalent ways of describing phenomena aren't always psychologically equivalent, they can lead to different types of new theories.

We can determine possible causal connections by evaluating the interval, (ct)² - x², separating pairs of events. If the interval is positive, then a causal connection is possible. This is known as a "timelike" interval. If it is negative ("spacelike") then no causal connection is possible.