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Chapter 2: Special Relativity - Coggle Diagram

- - - - Event
        
        An instantaneous physical situation or occurrence associated with a point in spacetime (a specific place and time)
      - Frame of Reference
        
        A set of coordinates by which an observer observes an event.
        
        Consists of spatial x-, y- and z-axes and a clock keeping time t
    - - of an event (e.g. event A in frame S) expressed as A(x, y, z, t)
      - Any measurement made on a coordinate system is a relative measurement
        
        Depends on choice of origin.
        
        If event A in another frame S', coordinates expressed as A(x', y', z', t')
    - - Their corresponding axes are parallel to each other (x // x', y // y', z // z')
      - Frame S' moves with constant velocity v relative to frame S along x and x' axes
      - The two origins (O and O') coincide at t = t' = 0
  - - - Assumption: Time and length are absolute/invariant, where they don't change in going from one frame to the other.
      - Everything is relative
        
        No such thing as absolute motion
        
        No preferred inertial frame to work in
        
        Led to the formulation of Einstein's First Postulate
    - - By rearranging terms, inverse transformation can be obtained:
        x = x' + vt'
        y = y'
        z = z'
        t = t'
      - For velocity
        
        Relates velocity of object in second frame of reference to the first.
  - - - Led physicists to search for the medium of propagation of em waves, widely known as luminiferous ether/aether
        
        Why does Galilean Transform fail?
        Einstein's two postulates of special relativity
        
        Speed of light in vacuum must be constant
        
        Laws of Physics are the same for all non-accelerating reference frames
  - - - Intentions
        
        Measure velocity of light in different directions relative to Earth.
        
        Determine velocity of Earth relative to ether
      - Used specially constructed interferometer to observe light interference
      - Result forced them to reject the null hypothesis
        i.e. No experimental evidence for the existence of ether.
      - Discovered to a high degree of precision that the velocity of light relative to Earth is the same in all directions
        
        Theory of ether rejected
        
        Em waves do not need a medium to propagate
- - - - Puts speed of light as an 'absolute' standard of measure as its speed is invariant across inertial frames
        
        GT and Newtonian mechanics cannot be true
  - - - Time interval between 2 events (in this example is the emission and reception of light)
        
        When measured in any inertial frame is always longer than or equal to the proper time
      - Proper time - time interval between two events (in this example is the emission and reception of light)
        
        Measured by a frame such that the two events occur at the same position in that frame
  - - - Relativity of simultaneity implies that length us also relative.
    - - Proper length of train L' measured in rest frame (frame S')
        
        position of 2 endpoints need not be measured simultaneously
      - Length of train L measured in other frame (frame S)
      - From time dilation,
        
        γ > 1, length contraction!
        
        Length measured in any frame is always shorter than the proper length
        
        Length perpendicular to the direction of motion is never contracted
        
        Violates first postulate of relativity
    - - Decide which frame the stated time interval (or length measurement) is measured with respect to
      - Decide whether the time interval (or length) is the proper time interval ∆t' (or proper length L')
        
        Proper time ∆t' is the time interval between the two events measured by a frame such that they occur at the same position in that frame
        
        Proper length L' is the length of an object measured in its rest frame
      - Identify the 2 frames of reference
      - Compute Lorentz factor (γ) and apply the time dilation/length contraction formula
- - - - Can be written mathematically as
        
        β << 1
        
        β → 0
        
        v << c
        
        γ → 1
        
        In this limit, LT becomes identical to GT
  - - - time not absolute, time is relative
        
        events that occur simultaneously to one observer may appear to happen at different times to another observer
        
        simultaneity is relative
        
        The relativity of simultaneity is the concept that distant simultaneity (whether 2 spatially separated events occur at the same time) is not absolute but depends on observer's reference frame.
  - - - γ depends on speed, not velocity in any axis
      - In the classical limit, u << c and v << c, above equation reduced to that in GT
- - - - Rest mass m_0 is the mass of an object at zero velocity, v = 0
      - Relativistic mass m is the mass of an object at velocity v
    - - Rapidly moving particle undergoes increase in mass
      - When object travels at high velocities, its energy is comprised of both rest and kinetic energy, hence mass increases at high velocities
        
        However, objects with mass can never attain v = c as it would require infinite energy to move an object with infinite relativistic mass, which is not possible.
  - - - Consider a body moving close to the speed of light, as it is moving as fast as it possibly can, any force acted on it exerts energy and momentum in the body. This momentum given by p=mv cannot possibly increase the speed of the body and hence increase of momentum results in an increase of mass.
        
        There are two equations that allow us to come to the conclusion that E = mc^2, where m is the mass gained by the body as a result of the increase in momentum.
        
        Body moves at v = c, distance d moved per unit time is c or near enough
        
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        In the unit time (t = 1s) in which the force f acts, mass increases by an amount ∆m
        
        This explanation only works for this particular scenario. More proof required to show that Einstein's equation holds for all cases
    - - For particle with mass m, whose v = 0 or v << c,
        
        Approximately equal to Newtonian p = mv when v << c, where
      - By Newton's Second Law
        
        ∆p is also known as impulse
        
        Force an acceleration do not relate linearly at relativistic speed, except if they have the same direction, where F is the proportional to acceleration and hence relate linearly
        
        Constant force does not cause constant acceleration
        
        If force and acceleration in same direction,
        
        When v << c and γ ≈ 1, expression reduces to the familiar Newtonian F = ma
        
        As objects speed increases, γ approaches infinity
        
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        When speed approaches c
        
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    - - Relativistic Kinetic Energy
        
        When v << c, above equation approaches the Newtonian expression K = (1/2)mv^2
        
        For a stationary particle,
        
        K = 0J
      - Rest Energy of an Object
      - Total Energy of an Object
        
        When v << c, the above equation approaches
      - Principle of Conservation of Mass and Energy
        
        In some physical phenomena, neither the sum of the rest masses of the particles nor the total energy (other than rest energy) is separately conserved
        
        In an isolated system, when the sum of the rest masses changes, there is always an equal and opposite change in the total energy other than the rest energy.
    - - Relativistic Energy-Momentum Relationship
        
        p: Relativistic momentum of the particle
        
        Can be applied to massless particles (photons), where m = 0
        
        Suggests the possible existence of particles that have energy and momentum even when they have zero rest mass (m_0 = 0)
      - Energy and Momentum of Massless Particles
        
        E = pc
        
        Massless particles exist: photon
      - Explain why it is more convenient to use eV, keV, MeV, GeV for units of energy, eV/c, keV/c, MeV/c, GeV/c for units of momentum and eV/c2, keV/c2, MeV/c2, GeV/c2 for units of mass.
        
        It is more convenient to know the energy equivalent of a particle's mass rather than the proper mass itself when it comes to calculations, and the prefixes (k, M, G), makes calculations easier as well as the numbers being dealt with won't be very big. Moreover, when dealing with calculations related to particles with charge, they would be much simpler as one wouldn't need to deal with the energy per unit charge of the charged particles as their charge usually comes in eV.