3. Spacetime and Special Relativity Flashcards
Space, Time and Motion in Newtonian Physics
Outline
- absolute Euclidian space
- absolute time agreed on by all observers
- absolute free motion of a particle in the absence of forces
- these three points lead to the concept of an inertial frame which describes the laws of physics
Geodesics in Euclidian Space
-straight lines
Relativity
-each type of relativity describes a way of relating different reference frames / observers
Galilean Relativity
Definition
-in all inertial reference frames, the laws of physics should agree
Galilean Relativity
Position
-have reference frame O and reference frame O’ moving along the x axis with relative velocity v such that at t’=t=0, O=O’
-Galilean transformations:
t = t’
y = y’
z = z’
x = x’ + vt
Galilean Relativity
Velocity
-differentiate the position Galilean transforms:
dy/dt = dy’/dt’
dz/dt = dz’/dt’
dx/dt = dx’/dt’ + v
Galilean Relativity
Speed of Light
-in the frame O: w = (dx/dt, dy/dt, dz/dt) -and in frame O' : w' = (dx'/dt', dy'/dt', dz'/dt') -then: w = w' + v -what if w'=c, speed of light -then: w = w' + v = c + v > c -for v>0
Space, Time and Motion in Special Relativity
Outline
- Einstein’s principle of relativity: “all laws of physics are the same in all inertial frames”
- speed of light principle: “the speed of light is the same in all inertial frames
Special Relativity
Position Derivation
-start with the most general possible linear transformation:
x’ = Ax + Bt
t’ = Cx + Dt
-where A, B, C and D are at most functions of v
-using conditions:
x’=0, x=vt
x=0, x’=-vt’
Special Relativity
Position in terms of Av and Ev
y' = y z' = z x' = Av (x - vt) t' = Av (Ev x + t)
Special Relativity
Lorentz Transformation Derivation
- suppose there is a third frame O’’
- O’ is moving with speed v with respect to O and O’’ moves with speed w with respect to O’
- find an expression for {x’‘,t’’} in terms of {x,t}
- all of these relations should have the same structure since they are the same transformation
Special Relativity
Lorenzt Transformations
y' = y z' = z x' = [x - vt] \ √[1 - v²/c²] t' = [-xv²/c² + t]\√[1 - v²/c²]
Minkowski Metric
-spacetime requires a non-Eulician metric e.g. Minkowski metric
gm = -c²dtxdt + dxxdx + dyxdy + dzxdz
Minkowski Metric and the Lorentz Transformation
-the Minkowski metric is invariant under the Lorentz transformation
Lorentz Factor Definition
γ = 1 / √[1 - v²/c²]
γ≥1
Lorentz Transforms in Terms of the Lorentz Factor
t = γ (t' + v/c² x') x = γ(x' + vt') y = y' z = z'
Special Relativity
Speed of Light
-consider a particle moving with speed w’=dx’/dt’ along x’ axis in frame O’ and w=dx/dt in frame O:
-using the Lorentz transforms to find expressions for dx’ and dt’
=>
w = [w’ + v] / [1 + w’ v/c²]
-suppose w’=c, then w=c
=>
-the Lorentz transforms are consistant with the speed of light principle
Time in Special Relativity
t=t’
-each inertial frame has its own time
Special Relativity
Relativity of Simultaneity and Temporal Order
-consider two events, suppose in frame O’ they are separated by time interval Δt’ and their x’ coordinates differ by Δx’
-then Δt and Δx in frame O can be determined using the Lorentz transforms
=>
Δt = γv/c² Δx’
-assuming v>0, this is >0 f0r Δx’>0, =0 for Δx’=0 and <0 for Δx’<0
-hence, the order of events is different in different inertial frames
Special Relativity
Time Dilation Effect
-consider a standard clock at rest in O’, denote τ as the time of this clock in O’ (the proper time since the clock is at rest in this reference frame)
-then for this clock, Δx’=0 and Δt’=Δτ
-using the Lorentz transforms,
Δt = γΔτ
-since γ≥1, Δt ≥ Δτ
-similarly, any standard clock at rest in O will appear to run slower when observed in frame O’
-the rate of a moving clock slows down compared to time in the inertial frame where the clock is observed
Special Relativity
Relativity of Spatial Order
-consider two events, suppose in frame O’ they are separated by time interval Δt’ and their x’ coordinates differ by Δx’
-then Δx in frame O is given by the Lorentz transform
=>
Δx = γ(Δx’ + vΔt’)
-suppose Δx’=0, then Δx=γvΔt’
-assuming v>0, this is >0 for Δt’>0, =0 for Δt’=0 and <0 for Δt’<0
=>
-spatial order of events is different in different inertial frames
Special Relativity
Length Contraction Effect
-consider a bar at rest in frame O, denote l as its length measured in this frame (proper length) and suppose it is aligned with the x axis:
l = xb - xa = Δx
-it doesn’t matter when the ends of the rod, xa and xb, are measured in frame O since the bar is at rest in that frame
-in frame O’ the bar is moving so the coordinates need to be measured simultaneously e.g. at Δt’=0:
l’ = xb’ - xa’ = Δx’
-using the Lorentz transforms:
Δx=γΔx’ or l’=l/γ
-since γ≥1, l’ ≤ l
=>
-the length contraction effect, in frame O’, the coordinate grid of frame O appears contracted along the x-axis
-each inertial frame has its own space / coordinate grid
Spacetime in Special Relativity
- special relativity insists that absolute space and absolute tie don’t exits, to describe any physical process an inertial frame must first be specified
- spacetime is absolute, whichever frame is used to measure space and time, the spacetime description will be the same
Spacetime Interval General Formula
ds² = -c²dt² + dx² + dy² + dz²
-or separating into space and time components:
ds² = -c²dt² + dl²
Normalising Minkowski Spacetime Coordinates
ds² = -c²dt² + dx² + dy² + dz² -time coordinate is not normalised, let: xo = ct x1 = x x2 = y x3 = z => ds² = -dxo² + dx1² + dx2² + dx3²
What are the three types of spacetime interval?
1) space-like
2) time-like
3) light-like
Describe the future and past light cones diagram
- the observer is at the centre
- a flat surface with the observer at the centre describes a hypersurface of simultaneity, a surface on which time is constant
- the vertical axis represents time
- the past light cone is a cone down the negative time axis
- the future light cone is a cone up the positive time axis
- the cones make a 45’ angle with the hypersurface
- light can only travel on the surface of the cone
- events are represented by points
- any event that has a chance of effecting you in the present lies in your past light cone, any event that you can affect lies in your future time cone
Space-Like Spacetime Intervals
ds² = -c²dt² + dl²
-if ds² > 0, there exists a frame where dt=0
=>
ds² = dl²
-this represents lines on the hypersurface of simultaneity
Time-Like Spacetime Intervals
ds² = -c²dt² + dl²
-if ds² < 0, there exists a frame where dl=0
=>
ds² = -c²dt²
Light-Like Spacetime Intervals
ds² = -c²dt² + dl² -if ds²=0 but dt=0, then: |dl/dt| = c -represents lines on the cone surface -two events that can be connected via a light signal e.g. two events in the life of a photon
