Week 9 & 10 Flashcards
Postulates of special relativity
Laws of physics are independent of the intertial frame of the observer
c in a vacuum is constant for all observers
Inhomogeneous electromagnetic wave equations
Show γ for time dilation
Where v is the speed that S’ is moving (relative to an intern S)
Then divide through by t’ to get time in frame t
ct =
Show length contraction
Length as measured from S has contracted in the moving frame S’
Relativistic addition of speed
Derived from Lorentz transformations with speed added
Where v is speed of train and u is speed of ball being thrown down the train in direction of movement whilst on it
Lorentz boost
Translation of the space time points along hyperbola (hyperbolic rotation analogous to rotating in circle)
Mixing ct and x, y or z
This is a transformation between 2 inertial frames of reference
Proper time? Where to derive it
-(ct)2 + x2 + y2 + z2 = -c2τ 2
Is invariant under Lorentz transformations
τ is proper time
Classes of vectors in Minkowski space
Diagram of a 2D subspace of Minkowski space (time and a position)
World line? Gradient of
World line is the curve resulting from plotting motion (in 1d) against ct
v is speed in x direction
Crossing light speed barrier
Consequently from special relativity objects can’t cross barrier
Objects with non zero rest mass can’t be accelerated to c
Extend the space time diagram to 4D
Light cone forms by extending into 3 d as a second spatial axis is added
In 4D this is still referred to as a light cone
Lorentz group
Significance of Lorentz group
Preserves Minkowski inner product
10 independent transformations of Lorentz groups
Contravariant vector ? Contravariant vector ?
Show that S2 can be taken from a covariant and Contravariant vector to I
Four displacement
Fundamental 4 vector which defines an event
By definition is Lorentz vector
Derive 4 velocity
Shorthand for 4 velocity
Show that 4 velocity is Lorentz Invariant
4 momentum
Derive 4 momentum in terms of E
Taylor expansion of P0 shows the 2nd term to be a multiple of KE and therefore we conclude the term is some form of energy
In particular E = P0c
Energy momentum relation
E2 = p2c2 + m2c4
Relativistic version of Newton’s second law
Where p is relativistic motion
p = γmv
