Light and Special Relativity

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26 Terms

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Inertial vs non-inertial frames of reference

A frame of reference - the perspective from which the motion of objects is observed and/or measured

Inertial frame of reference (FOR) – a non-accelerating frame of reference

Non-inertial FOR – an accelerating frame of reference

Example: A camera fixed to the ground is stationary and thus will capture a video from an inertial frame of reference.

Example: A camera falling down will capture a video from a non-inertial frame of reference since the camera itself is experiencing gravitational acceleration.

  • The observer is experiencing no net force

  • Observer is moving at a constant velocity or is at rest

  • Can tell if you are in an inertial FOR by consulting another external frame of reference

  • All of Newton’s laws of motion (Newtonian physics) are valid

  • You can conduct an experiment, and the results will be the same than if you did it stationary

  • Forces can exist within the FOR, but not on the FOR itself

  • Observer is experiencing a net force in the same direction as acceleration

  • Can tell if you are in a non-inertial if you feel a fictitious force

    Fictitious force – an imaginary force as a result of N1L

    E.g. being dragged to the side of a turning car → centrifugal force is fictitious

  • All of Newton’s laws of motion (Newtonian physics) are NOT valid

  • You can conduct an experiment, and the results will NOT be the same than if you did it stationary

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Pendulum Experiment

Experiment used to tell if you are inside an inertial or non-inertial frame of reference

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The Special Theory of Relativity

Published by Einstein in 1905

Postulate

1st postulate

2nd postulate

Description

The laws of physics are the same in all inertial frames of reference (Galilean principle of relativity)

The speed of light in vacuum has the same value for all observers, regardless of the state of motion of the observer or the source

Impact

  • No physics is difference between a FOR at rest or an FOR moving at a constant velocity  same experiences; if an experiment was conducted, the same results would be achieved

  • No absolute stationary FOR

  • All inertial FORs are equivalent

  • Cannot detect constant motion without consulting another external FOR

  • There is no aether

  • Time and space are relative → they were considered fixed quantities in classical physics

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Galileo’s Principle of Relativity

Prosed by Galileo in 1632

The mechanical laws of physics are the same for every inertial observer

  • Results of any experiment will be identical

  • There is no way to differentiate between a frame moving at constant speed and a frame at rest

    • Cannot be done without consulting another FOR (e.g. looking outside the window to see movement, feeling wind on your face, looking at a satellite)

      → e.g. If I am inside a boat, and I cannot look outside or feel the waves rocking, I cannot tell whether I am moving with a constant velocity or stationary

    • Means all inertial frames of reference are equivalent

    • It is possible to determine that one frame is moving relative to another, but it is impossible to determine which of them is at rest

    • There is no absolute stationary

  • Velocity is always defined relative to some external point

    • E.g. Our car’s velocities are measured relative to the Earth

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Historical context about the Relative Speed of Light

  • Throughout history Light’s speed was calculated to be 3x108m/s

  • People wanted to know what speed this was measured relative to

  • People assumed light required a medium to travel through, like other waves

    • Huygen thought light was a longitudinal wave that travelled through perfectly elastic particles that fill up space (the aether)

    • Maxwell showed light was EM, but they still didn’t know what medium it needed to propagate in

  • In the 19th century, scientists did experiments to determine what medium light needed to travel in

    • It was proposed that the aether had an absolute frame of reference

    • ∴ Light waves had a fixed velocity relative to the aether

  • The aether theory was not backed up by experimental evidence

    • E.g. Michelson-Morley experiment tried to find how light’s speed changed relative to earth, but had no results proving that light’s speed changed

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Einstein’s thought experiment for the relative speed of light

“Suppose I am sitting in a train travelling at the speed of light. If I hole a mirror in front of me, will I see my reflection?”

Possible answer 1: No

  • If the train was travelling at the speed of light, the photons reflected from his face would not reach the mirror in order to be reflected back

  • Since he cannot see his own reflection in the mirror, he would know the train was travelling at a constant speed (c), without consulting an external frame of reference

  • This violates the principle of relativity as an observer can detect their motion despite being in an inertial frame of reference

Possible answer 2: Yes

  • This means that light would travel at its normal speed relative to the train, and he could see his reflection

  • This does not violate the principle of relativity  as he would not know whether he is moving or at rest in the train

  • However, based on the addition of velocities, the light inside the train would be travelling twice its original speed (2c) for a stationary observer outside the train

  • This is impossible because nothing can travel faster than the speed of light (c), even light itself

Einstein concluded that the principle of relativity cannot be violated ∴ he can see his reflection.

  • Resolved this issue by stating that the speed of light is the same for any observer

  • Since: speed = distance ÷ time, the stationary observer and the moving observer must perceive distance and time differently → ratio changes but speed stays the same

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Michelson-Morely Experiment: purpose and result

Purpose of Experiment: to measure the velocity of the earth through the aether

  • By measuring the difference in the speed of light relative to the earth due to the Earth’s motion relative to the aether (called the aether wind)

  • Scientists used Newtonian relative motion to theorise that Earth's movement through the aether → caused the aether wind

Results of Experiment

  • When the interferometer was rotated, there was no change in the interference pattern observed → NULL RESULT

  • Disproved the existence of the aether wind

  • This supported Einstein’s theory that the speed of light is constant in all reference frames

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Michelson-Morely Experiment: method

Method of Experiment

  1. Directed a beam of light at an angle of 45° to a beam splitter (half-silvered, half transparent mirror)

  2. The beam splitter splits the beam by allowing half of it to pass through and the other half to be reflected

  3. They used a Michelson interferometer (an optical instrument that splits a light beam into two paths, reflects them back, and then recombines them to create an interference pattern)

    1. The reflected light travelled to meet mirror 1, then was reflected back onto the beam splitter and into the detector

    2. The light that passed through the beam splitter travelled to meet mirror 2, then was reflected back onto the beam splitter and into the detector

    3. When these light beams meet at the detector, they interfere to create interference fringes

  4. The interferometer is rotated by 90° and the interference pattern is monitored during the rotation

    1. The rotation would cause the speed of light in each arm to change due to the changing direction of velocity relative to the aether wind

    2. The speed of light changing means that the two beams would arrive at the detector at different times, and their relative phase would be different

    3. This would change the interference pattern during rotation

<p><strong>Method of Experiment</strong></p><ol><li><p>Directed a beam of light at an angle of 45° to a beam splitter (half-silvered, half transparent mirror)</p></li><li><p>The beam splitter splits the beam by allowing half of it to pass through and the other half to be reflected</p></li><li><p>They used a Michelson interferometer (an optical instrument that splits a light beam into two paths, reflects them back, and then recombines them to create an interference pattern)</p><ol><li><p>The reflected light travelled to meet mirror 1, then was reflected back onto the beam splitter and into the detector</p></li><li><p>The light that passed through the beam splitter travelled to meet mirror 2, then was reflected back onto the beam splitter and into the detector</p></li><li><p>When these light beams meet at the detector, they interfere to create interference fringes</p></li></ol></li><li><p>The interferometer is rotated by 90° and the interference pattern is monitored during the rotation</p><ol><li><p>The rotation would cause the speed of light in each arm to change due to the changing direction of velocity relative to the aether wind</p></li><li><p>The speed of light changing means that the two beams would arrive at the detector at different times, and their relative phase would be different</p></li><li><p>This would change the interference pattern during rotation</p></li></ol></li></ol><p></p>
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Time Dilation

the difference in elapsed time between two clocks that are in motion relative to each other

  • Observers in different inertial frames will measure different time intervals between a pair of events

  • A moving clock runs slower than an identical stationary clock

    • More second’s pass on a stationary clock than a moving clock according to your perspective

    • The faster a thing moves relative to the observer, the slower it looks from a “stationary” observer

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Thought exp for time dilation equation

  • Light must travel further for the observer outside than for the observer inside the train

  • According to the 2nd postulate of Special Relativity, the speed of light is the same for both observers

  • Since light travels a further distance for the observer outside the train, they measure a longer time than for the observer inside the train

  • Hence, the observer’s clock inside the train runs slow compared to the observer outside

<ul><li><p>Light must travel further for the observer outside than for the observer inside the train</p></li><li><p>According to the 2nd postulate of Special Relativity, the speed of light is the same for both observers</p></li><li><p>Since light travels a further distance for the observer outside the train, they measure a longer time than for the observer inside the train</p></li><li><p>Hence, the observer’s clock inside the train runs slow compared to the observer outside</p></li></ul><p></p>
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Proper vs Dilated time

Proper Time

Dilated Time

·      The shortest time possible for the event

·      Event occurs at the same special location

·      The observer is stationary relative to the event (the event happens in front of you) → same FOR

·      One clock needed to measure

·      Longer time

·      Events occur at different spatial locations

·      The observer sees the event moving → different FORs

·      Two clocks (one at each event) needed to measure time

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Length Contraction

the difference in lengths of an object measured by two observers that are in relative motion compared to each other

  • The length will be shorter if the object is moving relative to the observer making the measurement

    • The observer at rest with respect to the object measured is the proper length

    • The observer moving with a relative speed will observe a shorter length

  • Only contracts in the same plane as the movement

Example: Ball only experiences length contraction in the horizontal dimension (not vertical) since that is the direction of velocity

<p>the difference in lengths of an object measured by two observers that are in relative motion compared to each other</p><ul><li><p>The length will be shorter if the object is moving relative to the observer making the measurement</p><ul><li><p>The observer at rest with respect to the object measured is the proper length</p></li><li><p>The observer moving with a relative speed will observe a shorter length</p><p></p></li></ul></li><li><p>Only contracts in the same plane as the movement</p></li></ul><p>Example: Ball only experiences length contraction in the horizontal dimension (not vertical) since that is the direction of velocity</p><p></p>
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Proper vs Contracted length

Proper Length

Contracted Length

·      The largest length possible

·      If the object is stationary relative to the observer measuring it → Same FOR

·      Seen in one spatial location

·      No relative velocity between measurer and object

·      Shorter length

·      If the object is moving with a constant speed relative to the observer → different FORs

·      Seen in more than one spatial location

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Thought exp for Length Contraction equation

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Limitation of Special Relativity

  • These effects are negligible when a frame of reference is not moving at a relativistic speed.

  • Relativistic effects can occur in non-inertial frames of reference but in these scenarios, they are not only attributed to special relativity. Effects due to general relativity must be considered in non-inertial frames of reference. As a result, only relativistic effects in inertial frames of reference are entirely due to special relativity.

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Evidence for Time Dilation and Length Contraction: Muons

Muons – unstable elementary particle

  • Have a charge equal to an electron

  • Have a mass 207 times larger than an electron

  • Produced by the absorption of cosmic radiation in the high atmosphere

  • Have an average lifetime of 2.2 𝜇s when measured in a FOR at rest with respect to them

    • They decay to produce energy and other particles

  • Speed is around 0.998c

  • According to classical physics: could only travel a distance of 600m before they decay ∴ most should not reach earth

  • However, lots of muons are detected on earth due to time dilation and length contraction

FOR

Time taken for muon to fall

Length of fall

Muon

Proper time

Contracted length

Observer on earth

Dilated time

Proper length

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Evidence for Time Dilation and Length Contraction: particle accelerators

Lithium Ions

  • In 2014, an experiment was conducted to demonstrate time dilation using lithium ions travelling at 0.338c in a particle accelerator.

  • The time interval between excitation of electrons in lithium ions and their return to ground state was measured when lithium ions are travelling at 0.338c and at rest.

  • Physicists found that the interval was longer for moving lithium ions compared to those at rest, as measured by a stationary observer in the laboratory.

  • This difference in time was consistent with time dilation.

Increased lifetimes of short-lived particles

  • Unstable particles (like muons) are produced travelling at high speeds

  • Due to high speed of these particles, their lifetime is observed to be longer in the Earth’s frame of reference → we measure dilated times

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Evidence for Time Dilation and Length Contraction: atomic clocks

Hafele-Keating Experiment

  • Involved flying atomic clocks on aeroplanes to directly measure time dilation

    • One clock left on the ground at Washington DC

    • One clock was flown around the world eastwards (same direction of earth’s spin)

    • One clock was flown around the world westwards (opposite direction of earth’s spin)

  • Once all the clocks returned to earth, the times were compared

    • The time on each clock was different due to the different amounts of time dilation

    • Atomic clocks have very high precision → needed to even the smallest amounts of time dilation

  • Experiments agreed with predictions of relativity

<p>Hafele-Keating Experiment</p><ul><li><p>Involved flying atomic clocks on aeroplanes to directly measure time dilation</p><ul><li><p>One clock left on the ground at Washington DC</p></li><li><p>One clock was flown around the world <strong>eastwards</strong> (<strong>same</strong> direction of <strong>earth’s</strong> spin)</p></li><li><p>One clock was flown around the world <strong>westwards</strong> (<strong>opposite</strong> direction of earth’s spin)</p></li></ul></li><li><p>Once all the clocks returned to earth, the times were compared</p><ul><li><p>The time on each clock was different due to the different amounts of time dilation</p></li><li><p>Atomic clocks have very high precision → needed to even the smallest amounts of time dilation</p></li></ul></li><li><p>Experiments agreed with predictions of relativity</p></li></ul><p></p>
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Mass Dilation

mass of an object appears to increase as its speed approaches the speed of light

  • To a stationary observer, the mass of a particle travelling at relativistic velocity becomes heavier (dilated).

  • Mass is a measure of inertia

    • Inertia increases with velocity

    • The faster you go, the harder it is to accelerate

  • Mass-energy equivalence: If an object travels near speed c, doing work will not increase its speed, but its total mass → rest mass is a form of energy

<p>mass of an object appears to increase as its speed approaches the speed of light</p><ul><li><p>To a stationary observer, the mass of a particle travelling at relativistic velocity becomes heavier (dilated).</p></li><li><p>Mass is a measure of inertia</p><ul><li><p>Inertia increases with velocity</p></li><li><p>The faster you go, the harder it is to accelerate</p></li></ul></li><li><p>Mass-energy equivalence: If an object travels near speed c, doing work will not increase its speed, but its total mass → rest mass is a form of energy</p></li></ul><p></p>
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Relativistic Momentum

  • Defined in such a way that the conservation of momentum will hold in all inertial frames

  • Relativistic momentum increases with velocity because mass increases with velocity

  • Relativistic momentum is measured using a body’s rest mass (m0) so the increase in momentum is only due to the increase in the body’s velocity.

    • Mass is dilated to conserve momentum for two observers in different FORs

<ul><li><p>Defined in such a way that the conservation of momentum will hold in all inertial frames</p></li><li><p>Relativistic momentum increases with velocity because mass increases with velocity</p></li><li><p>Relativistic momentum is measured using a body’s rest mass (m0) so the increase in momentum is only due to the increase in the body’s velocity.</p><ul><li><p>Mass is dilated to conserve momentum for two observers in different FORs</p></li></ul></li></ul><p></p>
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Limiting Velocity

  • The limiting speed for any object with mass is the speed of light (c)

  • Space, gluons and photons don’t have mass → can travel at speed c or faster

  • The requirement for a particle to reach c is physically impossible and would break existing laws of physics

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Mass-Energy Equivalence

Einstein also proposed that mass and energy are interchangeable

  • Mass can be converted into energy and energy can be converted into mass

  • Each quantity does not need to be conserved in its own form as they can inter-convert.

  • Doing work on an object will accelerate it and increase its energy

    • At low speeds, mass stays the same and speed increases → Increase energy

    • At high speeds, the speed stays the same and the relativistic mass increases → increase in energy

Conservation of mass and energy

Rest mass is a type of energy (stored energy), it can be transformed

  • Mass can be converted into other types of energy (rest mass decreases)

    • e.g. Moving an electron closer to the positive plate, some of its mass is converted into KE

  • Other types of energy can be converted into rest mass (rest mass increases)

    • e.g. using particle accelerators to collide particles into each other  creates new particles with increased mass than both of the particles masses combined due to conversion if KE into mass

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E=mc²

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Mass-Energy Equivalence: Nuclear reactions in the Sun

  • Involve the rearrangement of protons and neutrons in the nucleus of atoms

  • Result in new nuclei being formed

    • Energy is absorbed if the proton and neutrons end up with more energy after the reaction

    • Energy is released of the protons and neutrons end up with less energy after the reaction

  • High temperatures are required

    • Nuclei need to get close to each other

    • Nuclei are positively charged

    • Temp provides energy (KE) to counteract the repulsive force between nuclei

  • Take place in the centre of stars (e.g. Sun)

Hydrogen Fusion

4 Hydrogen → Helium + energy

4 protons → alpha particle (2 protons + 2 neutrons) + energy

  • Combine four protons (hydrogen nuclei) to make one alpha particle (helium nucleus)

  • This releases energy

  • The proton-proton chain is the main hydrogen fusion sequence that occurs in the Sun

Proton-proton chain

  1. Two protons (2x 1H) collide to form deuterium (2H) and release a positron and a neutrino

  2. Deuterium (2H) collides with a proton (H) to form Helium-3 (3H) and releases a gamma ray

  3. Helium-3 then fuses with another helium-3 to form an alpha particle and releases two hydrogens

  4. This alpha particle is a helium atom

<ul><li><p>Involve the rearrangement of protons and neutrons in the nucleus of atoms</p></li><li><p>Result in new nuclei being formed</p><ul><li><p>Energy is absorbed if the proton and neutrons end up with more energy after the reaction</p></li><li><p>Energy is released of the protons and neutrons end up with less energy after the reaction</p></li></ul></li><li><p>High temperatures are required</p><ul><li><p>Nuclei need to get close to each other</p></li><li><p>Nuclei are positively charged</p></li><li><p>Temp provides energy (KE) to counteract the repulsive force between nuclei</p></li></ul></li><li><p>Take place in the centre of stars (e.g. Sun)</p></li></ul><p></p><p><strong>Hydrogen Fusion</strong></p><p>4 Hydrogen → Helium + energy</p><p>4 protons  → alpha particle (2 protons + 2 neutrons) + energy</p><ul><li><p>Combine four protons (hydrogen nuclei) to make one alpha particle (helium nucleus)</p></li><li><p>This releases energy</p></li><li><p>The proton-proton chain is the main hydrogen fusion sequence that occurs in the Sun</p></li></ul><p><strong>Proton-proton chain</strong></p><ol><li><p>Two protons (2x 1H) collide to form deuterium (2H) and release a positron and a neutrino</p></li><li><p>Deuterium (2H) collides with a proton (H) to form Helium-3 (3H) and releases a gamma ray</p></li><li><p>Helium-3 then fuses with another helium-3 to form an alpha particle and releases two hydrogens</p></li><li><p>This alpha particle is a helium atom  </p></li></ol><p></p>
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Mass-energy equivalence: particle–antiparticle interactions

Annihilation: Matter + Antimatter = energy

Two world consists of two groups of particles

  • Matter, consisting of:

    • Protons, neutrons, electrons  make atoms

    • Unstable particles (e.g. muons, tau) that exist for short times

  • Antimatter

    • Every matter particle has a corresponding antimatter particle

    • Matter and antimatter particles have the same mass but opposite charge

  • When an annihilation reaction (pair annihilation) occurs:

    • All the masses of both particles convert into energy

    • Energy is released as two gamma rays (to conserve momentum)

<p><strong>Annihilation: </strong>Matter + Antimatter = energy</p><p>Two world consists of two groups of particles</p><ul><li><p>Matter, consisting of:</p><ul><li><p>Protons, neutrons, electrons  make atoms</p></li><li><p>Unstable particles (e.g. muons, tau) that exist for short times</p></li></ul></li><li><p>Antimatter</p><ul><li><p>Every matter particle has a corresponding antimatter particle</p></li><li><p>Matter and antimatter particles have the same mass but opposite charge</p></li></ul></li><li><p>When an annihilation reaction (pair annihilation) occurs:</p><ul><li><p>All the masses of both particles convert into energy</p></li><li><p>Energy is released as two gamma rays (to conserve momentum)</p></li></ul></li></ul><p></p>
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Mass-energy equivalence: combustion of conventional fuel

  • Chemical reactions involve the rearrangement of electrons between different atoms

  • Result in the energy of electrons changing as new chemicals are formed

  • Energy can either be absorbed or released by chemical reactions  depends on whether electrons finish with higher / lower energy

  • Depending on the precision of the measurements, we can see this change  if measurements are imprecise, we will not see the mass change, since the energy released is so tiny

Combustion – chemical reactions where fuel is burnt (reacted with oxygen) to release energy

Fuel + oxygen → products + energy

e.g. Hydrogen + oxygen → water + energy

  • Conservation of energy requires energy on both sides of the equations to be the same

  • Since energy is released in the reaction, the mass of the products must be less than the combined mass of fuel and oxygen