What is Relativity? - SharpSchool

What is Relativity?Relating measurements in one reference frame to those in a different reference frame moving relative to the first

1905 - Einstein’s first paper on relativity, dealt with inertial reference frames (Special Relativity)

1915 - Einstein published theory that considered accelerated motion and its connection to gravity (General Relativity)

Special RelativityGR describes black holes, curved spacetime, and the evolution of the universe; very mathematical

SR deals with a “special case” case of motion - motion at a constant velocity (acceleration is zero)

SR is restricted to inertial reference frames - relative velocity is constant

Reference FramesInertial Reference Frame:

Reference FramesInertial Reference Frame:

A reference frame in which Newton’s first law is valid

Reference FramesWhich of these is an inertial reference frame (or a very good approximation)?

a. Your bedroomb. A car rolling down a steep hillc. A train coasting along a level trackd. A rocket being launchede. A roller coaster going over the top of a hillf. A skydiver falling at terminal speed

Standard Reference Frames S and S’

Galilean Transformations of PositionIf you know a position measured in one inertial reference frame, you can calculate the position that would be measured in any other inertial reference frame...

Suppose a firecracker explodes at time t. The experimenters in reference frame S determine that the explosion happened at position x. Similarly, the experimenters in S’ (which moves at a velocity v) find that the firecracker exploded at x’ in their reference frame. What is the relationship between x and x’?

Galilean Transformations of VelocityIf you know the velocity of a particle in one inertial reference frame, you can find the velocity that would be measured in any other inertial reference frame...

Suppose the experimenters in both reference frames now track the motion of an object by measuring its position at many instants of time. The experimenters in S find that the object’s velocity is u. During the same time interval Δt, the experimenters in S’ measure the velocity to be u’.

Galilean Transformations of VelocityUse u and u’ to represent the velocities of objects with respect to reference frames S and S’.

Find the relationship between u and u’ by taking the time derivatives of the position equations. (Recall: ux = dx/dt)

ExampleAn airplane is flying at speed 200 m/s with respect to the ground. Sound wave 1 is approaching the plane from the front, sound wave 2 is catching up from behind. Both waves travel at 340 m/s relative to the ground. What is the speed of each wave relative to the plane?

A simpler example...Ocean waves are approaching the beach at 10 m/s. A boat heading out to sea travels at 6 m/s. How fast are the waves moving in the boat’s reference frame?

Einstein’s Principle of Relativity

All the laws of physics are the same in all inertial reference frames.

Maxwell’s Contribution

● Maxwell’s equations are true in all inertial reference frames

● Maxwell’s equations predict that electromagnetic waves, including light, travel at speed c = 3 x 108 m/s

● Therefore, light travels at speed c in all inertial reference frames.

Implications

Implications● Recent experiments use unstable

elementary particles, mesons, that decay into high energy photons of light.

● Every experiment designed to compare the speed of light in different reference frames has found that light travels at speed c in every inertial reference frame, regardless of how the reference frames are moving with respect to each other.

ExampleUse a Galilean transformation to determine the bicycle’s velocity.

Example

Repeat your measurements but measure the velocity of the light wave as it travels from the tree to the lamppost.

ExampleRepeat your measurements but measure the velocity of the light wave as it travels from the tree to the lamppost. ● Δx’ differs from Δx ● u’ differs from u● BUT experimentally, u’ = u…● What does this tell us about our assumptions

regarding the nature of time?

Events and Measurements

Event: a physical activity that takes place at a definite point in space and a definite instant in time.Spacetime coordinates (x, y, z, t)

Measurements

The (x, y, z) coordinates of an event are determined by the intersection of the meter sticks closest to the event.The event’s time, t, is the time displayed on the clock nearest the event.

Stop and Think

A carpenter is working on a house two blocks away. You notice a slight delay between seeing the carpenter’s hammer hit the nail and and hearing the blow. At what time does the event “hammer hits nail” occur?a. at the instant you hear the blowb. at the instant you see the hammer hitc. very slightly before you see the hammer hitd. very slightly after you see the hammer hit

Synchronization of Clocks

Detection of light wave sent out from origin.

How long does it take for light to travel 300 m?

Finding the time of an eventExperimenter A in a reference frame S stands at the origin looking in the positive x-direction. Experimenter B stands at x = 900 m looking in the negative x-direction. A firecracker explodes somewhere between them. Experimenter B sees the light flash at t = 3.0 µs. Experimenter A sees the light flash at t = 4.0 µs. What are the spacetime coordinates of the explosion?

Finding the time of an event

SimultaneityWhen two events occurring at different positions take place at the same time.

An experimenter in reference frame S stands at the origin looking in the positive x-direction. At t = 3.0 µs she sees firecracker 1 explode at x = 600 m. A short time later, at t = 5.0 µs, she sees firecracker 2 explode at x = 1200 m. Are the two explosions simultaneous? If not, which firecracker exploded first?

Stop and ThinkA tree and pole are 3000 m apart. Each is suddenly hit by a bolt of lightning. Mark, who is standing at rest midway between the two, sees the two lightning bolts at the same instant in time. Nancy is at rest under the tree. Define event 1 to be “lightning strikes tree” and event 2 to be “lightning strikes pole.” For Nancy, does event 1 occur before, after, or at the same time as event 2?

A “Thought Experiment”...A long railroad car is traveling to the right with a velocity v. A firecracker is attached to each end of the car, just about the ground. Each firecracker will make a burn mark on the ground when where they explode. Ryan is standing on the ground; Peggy is standing in the exact center of the car with a light detector.

The Event in Ryan’s Frame

The Event in Peggy’s Frame

The real sequence of events in Peggy’s reference frame

Relativity of Simultaneity

Two events occurring simultaneously in reference frame S are not simultaneous in any reference frame S’ moving relative to S.

Stop and ThinkA tree and a pole are 3000 m apart. Each is hit by a bolt of lightning. Mark, who is standing at rest midway between the two, sees the two lightning bolts at the same instant of time. Nancy is flying her rocket at v = 0.5c in the direction from the tree toward the pole. The lightning hits the tree just as she passes by it. Define event 1 to be “lightning strikes tree” and event 2 to be “lightning strikes pole.” For Nancy, does event 1 occur before, after, or at the same time as event 2?

Time Dilation

Time is no longer an absolute quantity: it is not the same for two reference frames moving relative to each other.

● Time interval between two events● Whether two events are simultaneousDepends on the observer’s reference frame.

Time Dilation - A light clockThe light source emits a very short pulse of light that travels to the mirror and reflects back to a light detector next to the source. The clock advances one “tick” each time the detector receives a light pulse and the light source immediately emits the next light pulse.

Time Dilation - A light clockTwo experimenters measure the interval between two clicks of the light clock. The clock is at rest in reference frame S’. Reference frame S’ moves to the right with velocity v relative to reference frame S.

Time Dilation - A light clockEvent 1: the emission of a light pulseEvent 2: the detection of that light pulseIn frame S, Δt = t2 - t1 In frame S’, Δt’ = t’2 - t’1

Time Dilation - A light clockIn terms of h and c, what is the time interval, Δt’, in the clock’s rest frame, S’?

Compare this to the time interval, Δt, in reference frame S.(Use the classical analysis approach in which the speed of light does depend on the motion of the reference frame relative to the light source.)

Time Dilation - A light clock


Time Dilation - A light clockA classical analysis finds that the clock ticks at exactly the same rate in both frame S and frame S’.

Show that the time intervals are not the same according to the principle of relativity.


Time Dilation

Δτ is the time interval between two events that occur at the same position and called proper time.

Clocks moving relative to an observer are measured by that observer to run more slowly.(Or, the time interval between two ticks is the shortest in the reference frame in which the clock is at rest.)

Example - from the Sun to Saturn

Saturn is 1.43 x 1012 m from the sun. A rocket travels along a line from the sun to Saturn at a constant speed 0.9c relative to the solar system. How long does the journey take as measured by an experimenter on Earth? As measured by an astronaut on the rocket?

Stop and Think

Molly flies her rocket past Nick at constant velocity v. Molly and Nick both measure the time it takes the rocket, from nose to tail, to pass Nick. Which of the following is true?a. Both Molly and Nick measure the same amount of

time.b. Molly measures a shorter time interval than Nick.c. Nick measures a shorter time interval than Molly.

The Twin Paradox

George and Helen are twins. On their 25th birthday, Helen departs on a starship voyage to a distant star. Her starship accelerates almost instantly to a speed of 0.95c and that she travels 9.5 light years (9.5 ly) from Earth. Upon arriving, she discovers that the planets circling the star are inhabited by fierce aliens, so she immediately turns around and heads home at 0.95c.

The Twin Paradox

A light year is the distance that light travels in one year.According to George:● how old will he be when his sister returns?● how old will Helen be when she returns to Earth?According to Helen:● how old will she be when she returns to Earth?● how old will George be when she returns to Earth?

Reconciling the Twin Paradox

It is logically impossible for each to be younger than the other at the time they are reunited. ● Are the situations truly symmetrical? ● Do both observers spend the entire time in

an inertial reference frame?● Who is actually younger?

Length Contraction

Consider the rocket that traveled from the sun to Saturn in the previous example. What is the length of the spatial interval in both the S and S’ reference frames?

Length Contraction

Length Contraction

The distance between two objects in reference frame S’ is not the same as the distance between the same two objects in reference frame S.

Where l is the proper length.

The distance from the sun to Saturn

A rocket traveled along a line from the sun to Saturn at a constant speed of 0.9c relative to the solar system. The Saturn-to-sun distance was given as 1.43 x 1012 m. What is the distance between the sun and Saturn in the rocket’s reference frame?

Another Paradox?Vignesh and David are in their physics lab room. They each select a meter stick, lay the two side by side, and agree that the meter sticks are exactly the same length. They then go outside and run past each other, in opposite directions, at a relative speed v = 0.9c. ● Determine the length of each meter stick as

they move past one another.

Another Paradox? No!Relativity allows us to compare the same events as they’re measured in two different reference frames. The events by which Vignesh measures the length (in Vignesh’s frame) of David’s meter stick are not the same events as those by which David measures the length (in David’s frame) of Vignesh’s meter stick.

Another Paradox? No!If this weren’t the case, then we could tell which reference frame was “really” moving and which was “really” at rest. The principle of relativity doesn’t allow us to make that distinction.Each is moving relative to the other, so each should make the same measurement for the length of the other’s meter stick.

Quiz 24

Length Contraction at v << c

Using the length contraction equation, the length of a 1.00 m arrow (as measured at rest) is calculated to be 1.00 m when it moves at 300 m/s relative to an observer.

Isn’t length contraction supposed to make the measured length less than 1.00 m?

Binomial Expansion

Most calculators do not have the precision for this calculation. We can use binomial expansion to get around this limitation.

(1 - x)1/2 ≈ 1 - ½ x

What is the amount of length contraction for the arrow?

Simultaneity, Time Dilation, & Length ContractionChapter 37, Section 37-1 through 37-6p. 945#1-12

The Spacetime IntervalA firecracker explodes at the origin of an inertial reference frame. Then, 2.0 μs later, a second firecracker explodes 300 m away. Astronauts in a passing rocket measure the distance between the explosions to be 200 m. According to the astronauts, how much time elapses between the two explosions?

The Spacetime Interval

Show that

The Spacetime Interval

A firecracker explodes at the origin of an inertial reference frame. Then, 2.0 μs later, a second firecracker explodes 300 m away. Astronauts in a passing rocket measure the distance between the explosions to be 200 m. According to the astronauts, how much time elapses between the two explosions?

Everything is relative?

Time intervals & space intervals

Not spacetime intervals - the spacetime interval s between two events is not relative. It is agreed upon by experiments in inertial reference frames.

Stop and Think

Beth and Charles are at rest relative to each other. Anjay runs past at velocity v while holding a long pole parallel to his motion. Anjay, Beth, and Charles each measure the length of the pole at the instant Anjay passes Beth. Rank in order, from largest to smallest, the three lengths LA, LB, LC.

Lorentz Transformations

In classical relativity, t’ = t and the Galilean transformation lets us calculate the position of an event in frame S’.

Is there a similar transformation that lets us calculate an event’s spacetime coordinates in frame S’?


Transformation must:1. agree with Galilean transformations at v << c2. transform both space and time coordinates3. ensure that c is the same in all reference

frames


where ɣ is a dimensionless function of velocity that goes to 1 as velocity goes to 0.Event 1 - light is emitted from the origin of both reference framesEvent 2 - light strikes a light detector

Lorentz TransformationsEvent 1: x = x’ = 0 and t = t’ = 0Event 2: (x, t) and (x’, t’)

What is the position of event 2 in each reference frame?

Substitute these expressions into the transformation equations and solve for ɣ.


Derive the Lorentz transformations for t and t’

Hint: require x = x and transform a position from S to S’ and then back to S.


Peggy and Ryan revisitedPeggy is standing in the center of a long, flat railroad car that has firecrackers tied to both ends. The car moves past Ryan, who is standing on the ground, with velocity v = 0.8c. Flashes from the exploding firecrackers reach him simultaneously 1.0 μs after the instant that Peggy passes him, and he later finds burn marks on the track 300 m to either side of where he had been standing.

Peggy and Ryan revisiteda. According to Ryan, what is the distance between the

two explosions, and at what times do the explosions occur relative to the time that Peggy passes him?

Peggy and Ryan revisitedb. According to Peggy, what is the distance between the

two explosions, and at what times do the explosions occur relative to the time that Ryan passes her?


Peggy and Ryan revisitedRelative to Peggy, how far does Ryan move between the first explosion and the second explosion?

In Peggy’s Reference Frame

Check your results

Use the equation for length contraction to find the length that Ryan measures.

Show that Ryan’s and Peggy’s calculations of the spacetime interval agree.


Chapter 37, Section 37-8p. 945-946#13-16, 23

Barn & Ladder Paradox

Task: Put a 20 m ladder into a 10 m barn.

Rule: Both doors need to be shut at the same time.

LengthUse the Lorentz transformations of xR and xL to find l in terms of L and β.

Note: both measurements are simultaneous

Does your result agree with our previous equation for length contraction?

VelocityFind the Lorentz transformation u’ in frame S’ in terms of u, v, and c.

Recall: u’ = dx’/dt’

Check that your equation is consistent with the Galilean transformation u’ = u - v when v<<c.

Lorentz Velocity Transformation

A rocket flies past the earth at 0.90c. As it goes by, the rocket fires a bullet in the forward direction at 0.95c with respect to the rocket. What is the bullet’s speed with respect to the earth?

Lorentz Velocity Transformation

If the rocket fired a laser beam in the forward direction as it traveled past the earth at velocity v, what is the laser beam’s speed in the earth’s reference frame?

Relativistic Momentum

Write the equation for the momentum of a particle of mass m moving with velocity u = Δx/Δt using the time measured by the particle.

Relate this expression for p to the familiar Newtonian expression by using the time-dilation equation to relate proper time interval measured by the particle to the time interval Δt measured in frame S.

Relativistic MomentumElectrons in a particle accelerator reach a speed of 0.999c relative to the laboratory. One collision of an electron with a target produces a muon that moves forward with a speed of 0.95c relative to the laboratory. The muon mass is 1.90 x 10-28 kg. What is the muon’s momentum in the laboratory frame and in the frame of the electron beam?

Relativistic Momentum

Giancoli Chapter 37, Section 37-9p. 946Problem #’s 24-28

The Cosmic Speed Limit

Relate momentum to force and consider the implications of graph (a). Examine both Newtonian and relativistic cases.

The Cosmic Speed Limit

The speed c is a “cosmic speed limit” for material particles, or any causal influence.

A causal influence is any information that travels from A to B and allows A to cause B.

The Cosmic Speed LimitSuppose there exists some kind of causal influence that can travel at speed u > c.

Use Lorentz transformations to determine how events A and B appear in a reference frame S’ that travels at an ordinary speed v < c relative to frame S.

Relativistic Energy

A particle of mass m moves through a distance ∆x during a time interval ∆t.

Turn the spacetime interval into an expression involving momentum by multiplying by (m/∆τ)2.

Relativistic Energy

Recall the relationship between ∆t and the proper time, ∆τ.

Multiply both sides by c2 and determine a value for the invariant in a reference frame where the particle is at rest. Recall that s2 is an invariant that is the same in all inertial reference frames.

Relativistic EnergyRecall the relationship between ∆t and the proper time, ∆τ.

Multiply both sides by c2 and determine a value for the invariant in a reference frame where the particle is at rest. Recall that s2 is an invariant that is the same in all inertial reference frames.

Relativistic Energy: Making sense of the invariantIf experimenters in frames S and S’ both make measurements on the particle of mass m they will find...

Relativistic Energy: Making sense of the invariantIf experimenters in frames S and S’ both make measurements on the particle of mass m they will find...

Relativistic Energy

Use the binomial approximation expression for to find how mc2 behaves when u << c.

Relativistic Energy

Use the binomial approximation expression for to find how mc2 behaves when u << c.

Relativistic Energy

There is an inherent energy associated with mass!

Define the total energy E of a particle to be

E = mc2 = E0 + K

Relativistic Energy

the total energy consists of a rest energy, E0 = mc2

and a relativistic expression for the kinetic energy, K = ( - 1)mc2 = ( - 1)E0.

An Alternate Expression

Write a final version of the expression below in terms of energy and momentum.

Relating momentum to energyE2 – (pc)2 = E0

2

Where under a Lorentz transformation, E0 is an invariant with the same value mc2 in all inertial reference frames.

Stop to ThinkAn electron moves through the lab at 99% the speed of light. The lab reference frame is S and the electron’s reference frame is S’. In which reference frame is the electron’s rest mass larger?a. In frame S, the lab frameb. In frame S’, the electron’s framec. It is the same in both frames.

Ex Kinetic Energy and Total Energy

Calculate the rest energy and the kinetic energy of (a) a 100 g ball moving with a speed of 100 m/s and (b) an electron with a speed of 0.999c. Recall me = 9.11 x 10-31 kg.

Ex 37-7 Pion’s kinetic energy

A 0 meson (m = 2.4 x 10-28 kg) travels at a speed v = 0.8c. What is its kinetic energy? Compare to a classical calculation.

Ex 37-8 Energy from nuclear decayThe energy required or released in nuclear reactions and decays comes from a change in mass between the initial and final particles. In one type of radioactive decay, an atom of uranium (m = 232.03714 u) decays to an atom of thorium (m = 228.02873 u) plus an atom of helium (m = 4.00260 u) where the masses (always rest masses) given are in atomic mass units (1 u = 1.6605 x 10-27 kg). Calculate the energy released in this decay.

Ex 37-9 Mass change in a chem rxn

When two moles of hydrogen and one mole of oxygen react to form two moles of water, the energy released is 484 kJ. How much does the mass decrease in this reaction?

Relativistic Energy

Giancoli Chapter 37, Section 37-11p. 946Problem #’s 30, 31, 32, 34, 35, 37, 39, 40, 41, 43, 44, 45, 47 p. 947-948General Problems: 62, 65, 70, 76

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What is Relativity? - SharpSchool