E=mc squaredSomething out of nothing:

What we’ve learned…

2

21

'

cv

TT

−=

2

21

cv

LL p

−=

time dilation

length contraction

…that length and time are not absolute but depend on the point of view of the observer

…that the speed of light is absolute and is the same for all observers in all reference frames.

…that length and time are inextricably linked in order to keep the speed of light the same in all reference frames.

…that this relationship between length and time can be represented in a four dimensional space called “spacetime”.

…that we can transform the spacetime coordinates in one frame to another moving along the x axis at speed v using the following relations.

…that spacetime preserves the “distance” between two points.

)/(1

1where)(

222

2

cvx

c

vttzzyyvtxx

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=′=′=′−=′ γγγ

€

Δs( )2

= cΔt( )2

− Δx( )2

= c t2 − t1( )( )2

− x2 − x1( )2

€

Δs( )2

= Δ ′ s ( )2

…that from the spacetime coordinates we can derive a relationship between the velocity in one frame and the velocity in a another

We’ve skirted this issue of a speed limit

(v< or =c)

)/(1 2cvu

vuu

x

xx ′−

−′=

Okay, but what does this “speed” limit do to Newton’s laws of motion??

To the conservation of energy?

Constant force applied - leads to acceleration - spaceship goes faster. As it’s speed increases it begins to resist acceleration.

F=ma…if F is constant, does that mean the mass of the spaceship is increasing?

Is momentum really conserved?:

In a frame S:v v v=0

0)(before =−+= vmmvp 0after =p

In a frame S’ moving to the right at speed v:

01 =′v 2v′ V ′

First use relativistic velocity transformation: )/(1 2cvu

vuu

x

xx ′−

−′=

€

′ v 1 =v1 − v

1− (v1v /c 2)=

v − v

1− (v)(v) /c 2[ ]

= 0

€

′ v 2 =v2 − v

1− (v2v /c 2)=

−v − v

1− (−v)(v) /c 2[ ]

=−2v

1+ (v 2 /c 2)

€

V ' =V − v

1− (Vv /c 2)=

0 − v

1− (0)(v) /c 2[ ]

= −v

)/(1

222before cv

mvp

+−

=

mvp 2after −=

A gedankenexperiment:

AB

A

B

In the rest frame of A:A is at rest, B comes along from the right at a significant fraction of the speed of light, and in a glancing collision, imparts a tiny fraction of it’s momentum to A (so it hardly slows down) and A (relatively slowly) rolls off in a direction perpendicular to the incident momentum.

In the rest frame of B: B is at rest, A comes along from the left at a significant fraction of the speed of light, and in a glancing collision, imparts a tiny fraction of it’s momentum to B (so it hardly slows down) and B (relatively slowly) rolls off in a direction perpendicular to the incident momentum.

If velocity appears smaller by a factor of γ then the mass must appear larger by a factor of γ!

Energy apparently is being assimilated into the mass of the object…

The mass of an object is it’s own rest frame is called the proper mass of the object, or more often, the rest mass.

)/(1 22 cu

ump

−=

rr

Relativistic momentum:

€

F =dp

dt=

mdu

dt

⎛

⎝ ⎜

⎞

⎠ ⎟

1− (u2 /c 2)[ ]3

2

Acceleration of a particle decreases under the action of a constant force, as we observed it would at the beginning of the lecture.

Relativistic energy

€

W = Fdx =dp

dtx1

x2∫x1

x2∫ dx

The change in the kinetic energy of an object is equal to the net work done on the object.

Insert:

€

F =dp

dt=

mdu

dt

⎛

⎝ ⎜

⎞

⎠ ⎟

1− (u2 /c 2)[ ]3

2

Noting that dx=udt

€

W =m

du

dt

⎛

⎝ ⎜

⎞

⎠ ⎟

1− (u2 /c 2)[ ]3

2x1

x2∫ = mudu

1− (u2

c 2)[ ]

320

u

∫

2

2

2

2

)(1mc

cu

mcKEW −

−== Note that there is a term

that is independent of the speed…the rest energy!

This term comes from the lower edge of the integration interval, u=0, it had energy before it started to move!

"It followed from the special theory of relativity that mass and energy are both but different manifestations of the same thing -- a somewhat unfamiliar conception for the average mind. Furthermore, the equation E is equal to m c-squared, in which energy is put equal to mass, multiplied by the square of the velocity of light, showed that very small amounts of mass may be converted into a very large amount of energy and vice versa. The mass and energy were in fact equivalent, according to the formula mentioned before. This was demonstrated by Cockcroft and Walton in 1932, experimentally."

Mass Energy Equivalence

2

2

2

2

)(1mc

cu

mcKEW −

−==

The total energy therefore includes this “rest energy”:

2

2

2

2

)(1mc

cu

mcE γ=

−=

We found that an object has energy while at rest.

Let’s think of a classical analogy-you can convert potential energy into kinetic energy (i.e. gravitational potential when you pedal up a hill)-you can convert electrostatic potential into electric power (i.e. when you charge, then discharge a capacitor).

Does this suggest that energy can be converted into mass, or that high energies can make mass materialize?

In Paris in 1933, Irène and Frédéric Joliot-Curie took a photograph showing the conversion of energy into mass. A quantum of light, invisible here, carries energy up from beneath. In the middle it changes into mass -- two freshly created particles which curve away from each other.

The experimental proof….

A convenient energy unit, particularly for atomic and nuclear processes, is the energy given to an electron by accelerating it through 1 volt of electric potential difference. The work done on the charge is given by the charge times the voltage difference, which in this case is:

The abbreviation for electron volt is eV.

Room temperature thermal energy of a molecule............................……....0.04 eV

Visible light photons....................................................................................1.5-3.5 eV

Energy for the dissociation of an NaCl molecule into Na+ and Cl- ions:....4.2 eV

Ionization energy of atomic hydrogen ........................................................13.6 eV

The masses of elementary particles are frequently expressed in term of electron volts by making use of Einstein's famous equation , where m is the mass of the particle and c is the speed of light.

Real world modern physics!

The top quark has a mass of 175 GeV. (as much as a gold atom!) They are not stable, but decay almost instantly, so they cannot be found in nature. So how do we produce them?

You can accelerate a proton which has a mass of 1 GeV (only 1/175 the rest mass of the top) to a kinetic energy of 1000 GeV and slam it head on with an anti-proton-and you have enough energy to produce a top quark!

“It's as if two tennis balls collided and a bowling ball flew out… “

difference of ~208 MeV

Grave applications: the dawn of the nuclear age…

Is relativistic energy conserved?

Okay, now you’re convinced that relativistic energy is conserved, so here’s yet another paradox…if a light particle (photon) can turn into an electron and a photon and vice versa, how do you reconcile that with the fact that the photon is massless??

)/(1 22 cu

ump

−=

rr

We’ve convinced ourselves that relativistic momentum is conserved.

2

2

2

2

)(1mc

cu

mcE γ=

−=

afterafter pp =

22222 )(mccpE +=If momentum is conserved then energy must be conserved.

Tune in next time when we talk about….

…the quantum theory of light.