1. Gravity is very hard: Einstein’s General Theory of Relativity
Forces and Work
What Kinds of Forces?
What kinds of forces can we have in relativity?
Instantaneous action at a distance?
These are no good because instantaneous effects violate relativity
No problem, we already know about electric and magnetic fields
Electric force
Magnetic force Q q
Gravity is very hard: Einstein’s General Theory of Relativity

2. Newton’s Laws and Force
Newton’s Second Law in non-relativistic physics:
Relativity
Newton’s Third Law:
Conservation of momentum comes from Newton’s Third Law and One Version of Newton’s Second Law

3. Cyclotron Motion
Consider a charged particle moving in a magnetic field:
Magnetic field points out of the slide
Force keeps speed the same
 is constant
Centripetal acceleration: q
B in Tesla
R in meters
q in Coulombs
p in kgm/s

4. Work Take time derivative of the first one Solve for dE/dt
Substitute the second
Substitute the third
Rewrite the velocity
Integrate over time

5. Sample Problem 1.9
An electron (m = keV/c2) at rest is placed in an electric field of magnitude V/cm. How long does it take, and how far does it go, before it reaches a velocity of v = 0.500c?
Work formula:
Momentum formula:

6. Mass of composite object is not the sum of its parts
Composite Objects and Invariant Mass
Suppose I have a box containing two objects of mass m moving with equal and opposite velocities u. What does momentum and energy look like for this whole object? u u m m
Looks exactly like a particle at rest of mass M = 2m.
If you Lorentz boost this object, the Energy/momentum will transform exactly the same way as a single object.
If we can’t see inside this object, we can’t tell it’s not a single object with this mass.
Mass of composite object is not the sum of its parts

7. Finding the Invariant Mass
Suppose I have a box containing many objects of various masses, moving at various velocities. What mass object M can have the same momentum and energy as the whole mess? u1 m1 u3 u4 u2 m4 m2 m3
Effective Mass

8. Internal Energy and Heat
Suppose I have a (solid) containing many atoms. Now I heat it up. Does the mass change? Na Cl Na Cl Na Cl Cl Na Cl Na Cl Na
Heat
The atoms start to move around
This increases the energy E of each atom
But the total momentum is still 0
The total energy E of the whole object increases
The invariant mass of the whole object increases Na Cl Na Cl Na Cl Cl Na Cl Na Cl Na
Sample Problem 1.10
1 kg of water (specific heat J/g/C) is heated from 0C to 100C. How much does the mass increase?

9. Field Energy and Mass
Suppose I have a charged particle surrounded by electric fields. Does the field contribute to the mass? q
Electric fields have energy
Electric fields contribute to total energy
Electric fields contribute to total mass
The mass listed for a given particle includes this mass
Consider a hydrogen atom
Proton and electron have cancelling charges
Partly eliminates the electric field
Decreases total energy
Decreases invariant mass p e
Mass(H) < Mass(p) + Mass(e)
Binding energy counts like negative mass

10. Potential Energy
Any change in the potential energy of an object changes its mass
Heat
Electric energy
Chemical energy
Nuclear energy
[Gravitational energy is hard]
There is nothing particularly special about nuclear energy
Other types of energy are too little to significantly affect the mass
For nuclear energy, calculating the mass difference may be the easiest way to find the total energy produced.

11. Summary: Formulas You Need
Massless:

12. End of material for Test 1 Music and Lyrics by Herman Hupfield
Name That Song
This day and age we're living in Gives cause for apprehension With speed and new invention And things like fourth dimension.
Yet we get a trifle weary With Mr. Einstein's theory. So we must get down to earth at times Relax relieve the tension
And no matter what the progress Or what may yet be proved The simple facts of life are such They cannot be removed.
“When a man sits with a pretty girl for an hour, it seems like a minute. But let him sit on a hot stove for a minute and it's longer than any hour. That's relativity.”
Attributed to A. Einstein
End of material for Test 1
“As Time Goes By”
Music and Lyrics by Herman Hupfield