Simple Nature - Light and Matter

self-check D

Verify that this equation has the two properties we wanted..

Answer, p. 1058

KE compared tomc^2 at low speeds example 22

.An object is moving at ordinary nonrelativistic speeds. Compare

its kinetic energy to the energymc^2 it has purely because of its

mass.

.The speed of light is a very big number, somc^2 is a huge num-

ber of joules. The object has a gigantic amount of energy be-

cause of its mass, and only a relatively small amount of additional

kinetic energy because of its motion.

Another way of seeing this is that at low speeds,γis only a tiny

bit greater than 1, soEis only a tiny bit greater thanmc^2.

The correspondence principle for mass-energy example 23

.Show that the equationE=mγc^2 obeys the correspondence

principle.

.As we accelerate an object from rest, its mass-energy becomes

greater than its resting value. Nonrelativistically, we interpret this

excess mass-energy as the object’s kinetic energy,

K E=E(v)−E(v= 0)

=mγc^2 −mc^2

=m(γ−1)c^2.

Expressingγas

(

1 −v^2 /c^2

)− 1 / 2

and making use of the approx-

imation (1 +)p ≈1 +pfor small, we haveγ≈1 +v^2 / 2 c^2 ,

so

K E≈m(1 +

v^2

2 c^2

−1)c^2

=

1

2

mv^2 ,

which is the nonrelativistic expression. As demanded by the cor-

respondence principle, relativity agrees with newtonian physics at

speeds that are small compared to the speed of light.

7.3.3 ?The energy-momentum four-vector

Starting fromE=mγandp=mγv, a little algebra allows one

to prove the identity

m^2 =E^2 −p^2.

We can define an energy-momentum four-vector,

p= (E,px,py,pz),

and the relationm^2 =E^2 −p^2 then arises from the inner product

p·p. SinceEandpare separately conserved, the energy-momentum

four-vector is also conserved.

Section 7.3 Dynamics 437