2.1 The Apparent Need for Ether 2.2 The Michelson-Morley Experiment 2.3 Einstein’s Postulates 2.4 The Lorentz Transformation 2.5 Time Dilation and Length Contraction 2.6 Addition of Velocities 2.7 Experimental Verification 2.8 Twin Paradox 2.9 Space-time

2.10 Doppler Effect 2.11 Relativistic Momentum 2.12 Relativistic Energy 2.13 Computations in Modern Physics 2.14 Electromagnetism and Relativity

CHAPTER 2Special Theory of Relativity – Part 2

2.11: Relativistic Momentum

u u

u

u

00

x

y

p mu mup

00

x

y

pp mu mu

Now look at the same collision in moving frame (one moving with particle on the right).

u u

u

u

Using the relativistic velocity addition of

20

x

y

p mup

2 2

2In this case, moving frame velocity ; '1 /x x

uv u u u uu c

2 2

2So, before is '1 /

after is ' 2

x x

x x

mup muu c

p mu mu

Momentum is not conserved in moving frame

Linear momentum is not conserved if we use the conventions for momentum from classical physics even if we use the velocity transformation equations from the special theory of relativity.

There is no problem with the y direction, but there is a problem with the xdirection along the direction the ball is moving in each system.

Relativistic Momentum

Question　 How do we modify the definition of momentum

so that linear momentum is conserved in all frames?

Answer　 This accomplished by redefining momentum to be:

2 21 /mup muu c

However　 keep in mind that u in refers to the particle velocity, not the frame (v).

Frank (fixed or stationary system) is at rest in system K holding a ball of mass m. Frank throws his ball along his +y axis at speed u0.

Mary (moving system) holds a similar ball in system K that is moving in the x direction with velocity v with respect to system K.

Mary throws her ball along her –y’ axis at the same speed u0. The two balls collide and each of them catches their own balls as it rebounds.

Relativistic Momentum 2 21 /mup muu c

If we use the classical definition of momentum, the momentum of the ball thrown by Frank is entirely in the y direction

The change of momentum after collision as observed by Frank is

In system K according to Frank

0Fyp mu

02Fyp mu

0u

0u

Frank’s ball

In order to determine the velocity of Mary’s ball as measured by Frank we use the velocity transformation equations:

In system K according to Frank

Mary measures the initial velocity of her own ball to be

2

'1 ' /

MxMx

Mx

u vu vu v c

0' 0, 'Mx Myu u u

2 20

02

'1 /

1 ' /My

MyMx

u uu u v cu v c

0u

Mxu

Myu

Mary’s ball

Before the collision, the momentum of Mary’s ball as measured by Frank (the Fixed frame) becomes

Before

Before

For a perfectly elastic collision, the momentum after the collision is

After

After

The change in momentum of Mary’s ball according to Frank is

In system K according to Frank

Mary’s ball

Linear momentum is not conserved if we use the classical momentumeven if we use the velocity transformation equations from special relativity.

The total change in momentum of the collision, ∆pF + ∆pM, does not zero!

∆pF = ∆pFy = −2mu0

In system K according to Frank,

02 (1/ 1) 0F Mp p mu

Similarly, in system K’ according to Mary,

' ' 0F Mp p

Modification of the definition of linear momentum is required for preserving both linear momentum and Newton’s second law.

Relativistic Momentum

2 21 /mup muu c

keep in mind that u in refers to the particle velocity, not the frame (v).

Classical expression is accurate to within 1% as long as u < 0.14 c.

p mu

p mu

p mu

Relativistic Momentum2 21 /

mup muu c

[Example 2.9] Show that relativistic momentum is conserved in the above case.

For the Frank’s ball in system K according to Frank,

00 0 0 2 2

221 /

Fy

o

mup mu mu muu c

For the Mary’s ball in system K according to Frank,

2 2 2 2 2 2(1 / )M Mx My ou u u v u v c

Mxu v 2 200 1 /My

uu u v c

2 2

1 for the relativistic momentum: 1 /Mu c

2 2

0 02 2 2 2

0

2 1 / 221 / 1 /

My My My My

M

mu v c mup mu mu muu c u c

0Fy Fyp p p Same result for the system K’ according to Mary.

Wow! Same form!

2.12: Relativistic Energy

Relativistically:

Relativistic and Classical Kinetic Energies

212

K mu

2 ( 1)K mc

For speeds u << c,

2 2K mc mc

Total Energy and Rest Energy

Rewriting in the form

The term mc2 is called the rest energy and is denoted by E0.

This leaves the sum of the kinetic energy and rest energy to be interpreted as the total energy of the particle. The total energy is denoted by E and is given by

2 2K mc mc

The Equivalence of Mass and Energy

Example: “energy” stored in a stationary golf ball

Example: two blocks of wood that collide and stick together

We square this result, multiply by c2, and rearrange the result.

Relationship of Energy and Momentum

2 2 2 20E E p c 2 2 2 2 2Or, E m c p c

Massless Particles must have a speed equal to the speed of light c

Photons!

uu

2 2

11 /u c

2.13: Computations in Modern Physics

Electron Volt (eV)

Example: carbon-12Mass (12C atom)

Mass (12C atom)

Binding Energy

Binding Energy

Electromagnetism and Relativity Einstein was convinced that magnetic fields appeared as electric

fields observed in another inertial frame. That conclusion is the key to electromagnetism and relativity.

Einstein’s belief that Maxwell’s equations describe electromagnetism in any inertial frame was the key that led Einstein to the Lorentz transformations.

Maxwell’s assertion that all electromagnetic waves travel at the speed of light and Einstein’s postulate that light speed is invariant in all inertial framesseem intimately connected.

BF

EF

A moving charge on conducting Wire

According to wire frame, the force is magnetic.

According to moving charge (q0) frame, the force is electric.

BF

EF