Relativity and Gravitation Flashcards
two axioms
- all inertial reference frames are equivalent for the performance of all physical experiments
- the speed of light has the same constant value when measured in any inertial frame
events
things like a light flash or a bang that happen at a particular place and time
reference frames
coordinate systems used to describe where and when events happen.
examples of events
car crash
example of non-event
- concert as it has a duration - start of concert would be
- whole country clapping at once as no specific place
inertial reference frame
reference frame in which Newton’s first law holds
not accelerating
simultaneous
same time
same place
when we talk about the time of an event, we always mean…
the time of the event as measured on a clock carried by a local observer (same spatial position as the event)
how to measure the length of a moving object
have to measure both ends of the object simultaneously
position observers at strategic points in the reference frames of interest so coords known
observers make records of events which happen at their location and then compare with each other
how to measure length of a moving train
subtracting the coords of the two observers who observed opposite ends of the train at a prearranged time
if observers were inside the train this is a complicated way of getting the same value as a measuring tape
measuring lengths of moving objects - this approach relies on what 3 things?
- specific procedure for synchronising clocks
- assumes two events happening same time and place are simultaneous
- assume moving clocks measure the passage of time accurately
clock hypothesis
assumption that there is nothing intrinsic to motion or acceleration which stops a clock being a reliable measure of time
an accelerating clock will tick at the same rate as an non-accelerating clock
concrete example of a good clock
atomic clock
depends on fundamental physics to mark out a timescale
two inertial reference frames S and S’ with spatial coords (x,y,z) and (x’,y’,z’) and time coords t and t’ are in STANDARD CONFIGURATION if:
- they are aligned so that the (x,y,z) and (x’,y’,z’) axes are parallel.
- frame S’ is moving along x-axis with velocity v.
- zero of time coords coincide (t=t’=0) so origin of S’ frame is always at xs’=vt in frame S
galiliean transformation
relates two frame in standard config (wrong)
x’=x-vt
y’=y
z’=z
t’=t
differentiating galilean transformation
diff once:
v’x=vx-v
v’y=vy
v’z=vz
diff again:
a’=a
equations of motion in reference frames
replace unprimed terms with primed ones
exactly the same physics
what proved galilean transformation wrong
Maxwell’s equations
if gt true, physics different in moving frames
maxwell correct
RP correct
gt only approximate
special relativity needed
principle of relativity
all inertial frames are equivalent for the performance of all physical experiments
(includes chemistry,biology etc)
second axiom
speed of light is a universal constant
only case where two events are unambiguously simultaneous
if they take place at exactly the same points in space
light clock
idealised timekeeper
flash of light leaves a bulb, bounces off a mirror and returns (one tick)
2L=c delta t’
this gives lorentz factor
light clocks conclusion
both clocks are perfectly accurately measuring the passage of time; time is flowing differently for the two observers
lorentz factor graph
even at nearly 90% of c, gamma=2
beyond this exponential
infinite at v=c
natural units
time as a measure of distance
c=1, unitless number
light meter is time unit
meters as unit of time
factors of c disappear from eqns
which axis is vertical in Minkowski diagram
t
moving train of Minkkowki diagram
diagonal (moving through space with time)
front and back of train lines parallel as same v
Relationship between the two sets of coordinates in Minkowski diagram
x’=xcos theta+ y sin theta
y’=-x sin theta + y cos theta
distance r is same in both frames
invarient of the transformation
x^2+y^2 = r^2 = x’^2+y’^2
invarient interval, delta s^2
delta t^2 - delta x^2
Space like separated
Interval -ve
Time like separated
Interval +ve
Light like separated
Interval 0
Relativity of simultaneity
Clocks will remain synchronised only in their own frame
momentum 4-vector
P=mU=m gamma(1,v)
length on momentum 4-vector in the rest frame of the particle
m which points along the particle’s world-line (direction of particle’s movement)
pair of incoming particles which collide and produce set of outgoing particles
P1+P2=P3+P4
m1γ(v1)+m2γ(v2)=m3γ(v3)+m4γ(v4)
as v approaches 0, γ(v)
approaches 1
low-speed limit of the spatial part of the vector P is
just mv
ie momentum conserved
time component of the 4-momentum
P^0=γm
low v limit of time component of the 4-momentum
from taylor series for (1+x)^n
γ=(1-v^2)^-1/2
=1+v^2/2 + o(v^4)
p^0=m+1/2mv^2+0(v^4)
o(v^4)
order of v^4
what is p^0
relativistic energy of a particle with mass m and velocity v
E=P^0=γm ie E=γmc^2
magnitude of 4-vector will be frame-invariant so
will express something fundamental about the vector, analogous to its length
P.P=
m^2
why we cannot define a 4-velocity vector for a photon by the same route as before
dR.dR is always zero and so proper time is also zero
alternative route for 4-velocity vector for a photon
P^0 component is related to energy so can define 4-momentum for massless particle
velocity-4vector is null so 4-momentum also null so P.P=0, p.p=E^2
E^2=p^2
P^1_2 notation
component 1(x-component) of the vector P2
superscripts denote different momentum components
subscripts denote different particle momenta
conserved
refers to a quantity which is not changed by some process
this refers to one frame at a time and a conserved quantity will typically have different numerical values in different frames
covariant
refers to an equation which does not change its form when it is transformed form one frame to another
invariant
refers to a quantity which is not changed by some transformation
this necessarily refers to more than one reference frame
constant
refers to a quantity which does not change in time
what can be considered conserved, invariant and constant
speed of light
centre of momentum frame
frame in which the total spatial momentum is zero so that incoming particles have equal and opposite spatial momenta
centre of mass energy
energy available for particle production in the centre of momentum frame
in general relativity it is not mass that is the source of gravitation but
4-momentum
compton scattering
an incoming photon strikes a stationary electron and both recoil
in the context of GR, an inertial frame is
a frame in which SR applies and thus the frame in which the laws of nature take their corresponding simple form
the equivalence principle
uniform gravitational fields are equivalent to frames that accelerate uniformly relative to inertial frames
equivalence principle version 2
all local, freely falling non-rotating laboratories are fully equivalent for the performance of physical experiments
the non-free-fall observer will measure light path of shining torch
as being curved in the gravitational field
ie even massless light is affected by gravity
tidal effects
2 objects will fall towards centre of the Earth
curvature of space is 1/R and as R gets larger
surface becomes locally more like flat euclidean space (curvature tends to zero)
hyperbolic paraboloid
surface with constant negative curvature
internal angles of triangles add up to less than pi, circumference >piD
metric of Minkowski space
ds^2=dt^2-dx^2-dy^2-dz^2
geodasics
shortest distance between two points in the space (works for euclidean space and great circles on sphere)
longest distance between two points in Minkowski space - path of extremal length
equivalence principle, version 3
any physical law that can be expressed geometrically in SR has exactly the same form in a locally inertial frame of a curved spacetime
spacetime tells matter…
how to move
tensor
more general geometrical object to describe energy-momentum of an extended object
‘vectors squared’
einstein tensor, G
specific tensor constructed in terms of second derivatives of the metric and therefore characterises curvature of spacetime that metric describes
einstein’s equation
equation for gravity in which the constant k plays same broad role as G in Newton’s theory
G=-kT
einstein’s equation represents a
sequence of constraints: the distribution of energy and matter, represented by the energy-momentum tensor T, constraints G which in turn constrains the metric of the space
matter tells spacetime
how to curve
if gravitating mass is small such as something around a planet this creates
the ‘weak-field’ approximation to the solution
at speeds much less than that of light, near a ‘small mass’, the spacetime
is approximately that of empty space (approximately Minkowski space)
a gravitational time dilation
a clock at a low altitude observed by an observer at a higher one is measured as slower than equivalent clock beside the observer
what are the geodesics of minkowski spacetime
the r(t) that the geodasics traces out is the same as the motion of a test particle in a gravitational field
Schwarzschild solution
metric in terms of the coordinates (t,r,theta, X) where theta and chi are the usual polar coords
Schwarzschild radius
R=2GM
asymptotically flat
at large r it reduces to the Minkowski metric
why coefficient dr^2 is divergent (appears to be a singularity)
coordinate system goes wrong here
the schwarzschild radius is the
radius of the event horizon
the spacetime when the central mass is physically compact enough that it is entirely contained within 2GM is the spacetime of a black hole
schwarzschild solution includes some geodesics which spiral around the t-axis as
near ellipses
to find a metric for the universe as a whole, we start with the observation that
on the largest scales the universe appears homogeneous and isotropic (same at all spatial points and spatial directions)
this is the copernican principle
cosmological constant
scaling factor Λ
