Special Relativity

PhysicsMontwood High School

R. Casao

Motion is Relative• Whenever we talk about motion, we must always specify the

point from which motion is being observed and measured.• The place from which motion is observed and measured is the

frame of reference.• A coordinate system moving at a constant velocity is called an

inertial frame of reference.• An object may have different velocities relative to different

frames of reference.– To measure the speed of an object, we first choose a frame

of reference and pretend that we are at rest in that frame of reference.

– Then we measure the speed with which the object moves relative to us

• For example: a person walking along the aisle of a moving train may be walking at a speed of 1 km/hr relative to his seat but at 60 km/hr relative to the train station (at rest on the ground).

Michelson-Morley Experiiment• Michelson and Morley attempted to measure the motion of

the Earth through space.• Because light travels in waves, it was assumed that something

in space , called ether, filled all of space and was a frame of reference attached to space.

• An interferometer was used to split a beam of light from a source into two beams at right angles to each other.

• The beams are then reflected and recombined to show whether there was any difference in average speed over the two back-and-forth paths.

• The interferometer was set with one path parallel to the motion of the Earth in its orbit and Michelson or Morley watched for any changes in average speed as the apparatus was rotated to put the other path parallel to the motion of the Earth.

• The interferometer was sensitive enough to measure the difference in the round-trip times of light going with and against Earth’s orbital velocity of 30 km/s and going back and forth against Earth’s path through space.

• No changes in average speed were observed; the speed of light was still c, not c + the motion of the source or receiver or c – the motion of the source or receiver.

• Irish physicist Fitzgerald suggested that the length of the experimental apparatus shrank in the direction in which it was moving by an amount to counteract the suggested variation in the speed of light.

• The shrinkage factor was worked out by Dutch physicist Lorentz and was:

• Einstein showed that was the shrinkage factor of space itself.

• It isn’t known how much the Michelson-Morley experiment influenced Einstein, but Einstein advanced the idea that the speed of light in free space is the same in all reference frames.

• The idea was against the classical ideas of space and time.• Speed is a ratio of distance through space to the time required

to do so.• For the speed of light to be constant, the classical idea that

space and time are independent of each other had to be rejected.

• Einstein saw that space and time are linked.

2

21 vc

2

21 vc

• Einstein saw all motion as relative to arbitrary frames of reference.

• A passenger on a train who looks out the window sees the train on the next track moving by the window.

• The passenger is aware of the relative motion between their train and the other train and cannot tell which train is moving.

• The passenger may be at rest relative to the ground and the other train may be moving, or the passenger may be moving relative to the ground and the other train is at rest, or both trains could be moving relative to the ground.

• If you were in a train with no windows, there would be no way to determine whether the train was moving with uniform velocity (a = 0 m/s2) or at rest (a = 0 m/s2 too).

• Einstein’s first postulate of the special theory of relativity is: All laws of nature are the same in all uniformly moving frames of reference (a = 0 m/s2).

• On a jet airplane going 700 km/hr, coffee pours as it does when the plane is at rest; a jumping Mr. Murillo on an airplane lands in the same spot as he would in the plane at rest on the ground; if we swing a pendulum in the moving plane, it swings as it would if the plane were at rest on the ground.

• The laws of physics within the moving airplane are the same as those in the plane at rest. No experiment can be devised to detect uniform motion, only accelerated motion.

• Einstein wondered what a light beam would look like if you traveled alongside it.– According to classical physics, the light beam would be at rest to such

an observer moving at the speed of light.– Einstein became convinced that no one could move with a light beam

and concluded that no matter how fast two observers moved relative to each other, each of them would measure the speed of light to be 3 x 108 m/s.

• Einstein’s second postulate of the special theory of relativity is: the speed of light in free space has the same measured value for all observers, regardless of the motion of the source or the motion of the observer; the speed of light is constant.

Simultaneity• We say that two events are simultaneous if they occur at the

same time.• Consider a light source in the exact center of a rocket ship

– When the light is switched on, light spreads out in all directions at speed c.

– Because the light source is equal distance from the front and back ends of the rocket ship, an observer inside the rocket ship finds that light reaches the front end at the same instant it reaches the back end.

– This happens if the rocket ship is at rest or moving at constant velocity.– The events of hitting the back end and hitting the front end occur

simultaneously for the observer in the rocket ship.

• For an observer who views the same two events from another frame of reference, such as on the Earth as the rocket ship moves by, the two events are not simultaneous.– As light travels out from the source, the Earth observer sees the ship

move forward, to the back of the rocket ship moves toward the light beam while the front of the rocket ship moves away from the light beam.

– The light beam going to the back of the rocket ship has a shorter distance to travel than the light beam going forward.

– Since the speed of light is the same in both directions, the Earth observer sees the light beam hit the back of the rocket ship before seeing the light beam hit

the front of the rocket ship.• Two events that are simultaneous in one frame of reference need not be simultaneous in a frame moving relative to the first frame of reference.

Spacetime• The space we live in is 3-dimensional and we can specify the

position of any location in space with 3 dimensions.• Three numbers, the distance along the x-axis, the distance along

the y-axis, and the distance along the z-axis, will specify the position of a point in space.

• For example, a box is described by its length, width, and height.• The 3 dimensions do not give a complete picture because there

is a 4th dimension – time.• The box was not always a box of length, width, and height.

– It became a box only at a certain point in time on the day it was made.– It won’t always be a box because at any moment it may be crushed, burned, or torn apart.– The 3 dimensions of space are a valid description of the box only during a certain specified period of time.

• Everything exists in spacetime; everything exists in “the spacetime continuum”.

• Two side-by-side observers at rest relative to each other share the same reference frame and measurements of space and time intervals between events would be the same for both, so we say they share the same spacetime.

• If there is relative motion between them, the observers will not agree on measurements of space and time.– At ordinary speeds, the differences in their measurements are small.– At speeds near the speed of light, called relativistic speeds, the differences

in space and time differ in such a way that each observer will always measure the same ratio of space and time for light; the greater the measured distance in space, the greater the measured time interval.

– The constant ratio of space and time for light, c, is the unifying factor between different realms of spacetime.

Time Dilation• Imagine that we are able to observe a flash of light bouncing

back and forth between a pair of parallel mirrors.• If the distance between the mirrors is fixed, the arrangement is

called a light clock, and the back and forth trips for the flash of light take equal time intervals.

• An observer who travels along with the ship and watches the light clock sees the flash reflecting straight up and down between the two mirrors, just is would if the spaceship were at rest.

• Because the observer in the ship is moving along with the light clock, there is no relative motion between the observer and the clock; the observer and the clock share the same reference frame in spacetime.

• If we are standing on the ground as the spaceship passes us at 0.5·c, we do not see the light path as a simple up-and-down motion.

• Each light flash moves horizontally while it moves vertically between the two mirrors and we see the flash follow a diagonal path.

• From the Earth frame of reference, the light flash travels a longer distance as it makes one round trip between the mirrors.

• Because the speed of light is the same in all reference frames, the light flash must travel for a corresponding longer time between the mirrors in our Earth reference frame than it does in the reference frame of the onboard observer.

• From the definition of speed = distance/time, the longer diagonal distance must be divided by a correspondingly longer time interval to give the same value for the speed of light.

• This stretching out of time is called time dilation.• All clocks run more slowly when moving than when at rest.• Time dilation has to do with the nature of time itself and not with the mechanics of clocks.

• The relationship between the time to (called proper time) in the frame of reference moving with the clock and the time t measured in another frame of reference (called relative time) is:

where v represents the speed of the clock relative to the outside observer

• Proper time to is the time it takes for the light flash to move between the mirrors in a straight up-and-down motion (the frame of reference of the light clock).

2

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• The time t represents the time it takes the light flash to move from one mirror to the other as measured from a frame of reference in which the light clock moves with speed v.– Because the speed of the light flash is c and the time it

takes to go from position 1 to position 2 is t, the diagonal distance traveled is c·t.

– During this time t, the light clock traveled horizontally at speed v and moves a horizontal distance v·t from position 1 to position 2.

– The third side of the right triangle is the distance the light travels vertically up-and-down, c·to.

– Applying the Pythagorean theorem to the triangle, we get:

2 2 2

2 2 2 2 2 2

2 2 2 2 2 2

2 2 2 2 2

2

2 2 22 2

2 2

2 22 2

2 2

22 2

2

22

2

2

2

2 2

2 2

1

1

1 1

o

o

o

o

o

o

o

o

o o

c t v t c t

c t v t c t

c t c t v t

c t t c v

divide through by c

t c vc tc c

c vt tc c

vt tc

ttvc

t ttv vc c

• The quantity is the same factor used by Lorentz to explain the length contraction.

• The inverse of this quantity is called the Lorentz factor γ :

• This allows time dilation to be expressed as: t = γ·to

• the measurements made in one realm of spacetime need not agree with the measurements in another realm of spacetime; but the one measurement that all observers agree on is the speed of light, c.

• Time dilation has been confirmed in the laboratory with particle accelerators where the lifetimes of fast-moving particles increases as the speed goes up.

2

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2

2

1

1 vc

• Atomic clocks orbit the Earth as part of the Global Positioning System (GPS) and must adjust for the effects of time dilation in order to use signals from the clocks to pinpoint locations on Earth.

Addition of Velocities• For everyday objects in uniform motion (a = 0 m/s2) we combine

velocities by the simple rule: V = v1 + v2

• This rule does not apply to light, which always has the same velocity c.

• No matter what the relative velocities are between two reference frames, light moving at c in one frame will be seen to be moving at c in any other frame.

• If you try chasing light, you can never catch it.• No material object can travel as fast or faster than light (altho

that may be questionable these days).

Length Contraction• As objects move through spacetime, space as well as time

changes.• Space is contracted, making the objects look shorter when they

move by us at relativistic speeds.• Length contraction was first proposed by Fitzgerald and

mathematically expressed by Lorentz, but it was Einstein who saw that space itself contracts.

• Because Einstein’s formula is the same as Lorentz’s, the effect is called the Lorentz contraction:

where v is the relative velocity between the observed object and

theobserver, c is the speed of light, L

is the measured length of the

moving object, and Lo is the measured length of the object at rest.

2

21ovL Lc

• According to the Lorentz contraction equation, if an object were to somehow move at speed c, its length would be 0.

• This is one reason we say that the speed of light is the upper limit for speed of any moving object.

• Length contraction only takes place in the direction of motion. – If an object is moving horizontally, no contraction takes place vertically.– In the figure below, as the speed increases horizontally, length in the

direction of motion decreases.

2

21ovL Lc

Length Contraction

Bob’s reference frame:

The distance measured by the spacecraft is shorter.

Sally’s reference frame:

Sally

Bob

0

0

LLvt t

The relative speed v is the same for both observers:

220

/1 cv

tt

220 /1 cvLL

Relativisitic Momentum• The change inmomentum m·v of an object is equal to the

impulse F·t applied to it: F·t = Δm·v.– If we apply more impulse to an object that is free to move, the object

gains more momentum.– Double the impulse and double the momentum.– Does this mean that momentum can increase without limit? Yes.– Does this mean that speed can increase without limit? No; nature’s

speed limit for material objects is c.• For Newton, infinite momentum would mean infinite mass or

infinite speed.• Einstein showed that a new definition of momentum is required:

where p is momentum and γ is the Lorentz factor.

• This generalized definition of momentum is valid in all uniformly moving reference frames.

p m v

• For everyday speeds, γ is nearly equal to 1, so momentum is nearly equal to m·v. At normal speeds, Newton’s p = m·v is valid.

• At relativistic speeds, γ grows quickly and so does relativistic momentum.– As speed approaches c, γ approaches infinity. No matter how close to c

an object is pushed, it would still require infinite impulse to give it the speed needed to reach c.

– No body with mass can be pushed to the speed of light, or beyond it.• Subatomic particles are routinely accelerated to nearly the

speed of light.– Classically, the particles behave as if their masses increase with speed.– Einstein keeps mass constant, so it is γ that changes with speed.– The increased momentum of a high-speed particle is evident in the

increased “stiffness” of its trajectory.– The more momentum it has, the “stiffer” is its trajectory and the harder it

is to deflect.

• Charged particles moving in a magnetic field experience a force that deflects the particles from their normal paths.– For small momentum, the paths curve sharply.– For large momentum, there is a greater stiffness and the path curves only

a little.– This stiffness must be compensated for in circular particle accelerators

like cyclotrons and synchrotrons, where momentum dictates the radius of curvature.

– Momentum changes do not produce deviations from the straight-line path for particles traveling in a linear accelerator.

Mass, Energy, and E = m·c2

• Einstein also linked mass and energy.• Matter, even at rest and not interacting with anything, has an

“energy of being” called its rest energy.• Einstein concluded that it takes energy to make mass and that

energy is released if mass disappears.

• The amount of energy E is related to the amount of mass m by:E = m·c2

• The c2 is the conversion factor between energy units and mass units.

• Because of the large magnitude of c, a small mass corresponds to an enormous amount of energy.

• In chemistry, you learned that tiny decreases of nuclear mass in both nuclear fission and nuclear fusion produced enormous amounts of energy, all in accord with E = m·c2.

• A change in energy of any object at rest is accompanied by a change in its mass, usually too small to be measured.– The filament of a light bulb energized with electricity has more mass than

when it is turned off.– A hot cup of tea has more mass than the same cup of tea when cold.– A wound-up spring clock has more mass than the same clock when

unwound.

• The quantity c2 is:– a conversion factor that converts the measurement of mass to the

measurement of equivalent energy.– or it is the ratio of rest energy to mass:

• The c2 in either form of the equation has nothing to do with light and nothing to do with motion.– The magnitude of c2 = 9 x 1016 J/kg.– 1 kg of mass has energy of 9 x 1016 J.

• The equation E = m·c2 is more than a formula for the conversion of mass into other kinds of energy, it states that energy and mass are the same thing.

• If you want to know how much energy is in a system, measure its mass. For an object at rest, its energy is its mass.

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