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 Qq Gravity is very hard: Einstein’s General Theory of Relativity Magnetic forceElectric force

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

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

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

Sample Problem An electron (m = keV/c 2 ) 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:

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? m u m u 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

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? m1m1 u1u1 m2m2 u2u2 m3m3 u3u3 m4m4 u4u4 Effective Mass

Internal Energy and Heat Suppose I have a (solid) containing many atoms. Now I heat it up. Does the mass change? Na 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 ClNaClNaCl NaClNaClNa ClNaClNaCl NaClNaClNa Heat 1 kg of water (specific heat J/g/  C) is heated from 0  C to 100  C. How much does the mass increase?

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

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.

Massless: Summary: Formulas you need

End of material for Test 1 What’s the name of this 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. “As Time Goes By” Music and Lyrics by Herman Hupfield “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