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Astronomy 2 > Relativity and Gravitation > Flashcards

Relativity and Gravitation Flashcards

1
Q

two axioms

A
  1. all inertial reference frames are equivalent for the performance of all physical experiments
  2. the speed of light has the same constant value when measured in any inertial frame
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2
Q

events

A

things like a light flash or a bang that happen at a particular place and time

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3
Q

reference frames

A

coordinate systems used to describe where and when events happen.

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4
Q

examples of events

A

car crash

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5
Q

example of non-event

A
  1. concert as it has a duration - start of concert would be
  2. whole country clapping at once as no specific place
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6
Q

inertial reference frame

A

reference frame in which Newton’s first law holds

not accelerating

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7
Q

simultaneous

A

same time
same place

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8
Q

when we talk about the time of an event, we always mean…

A

the time of the event as measured on a clock carried by a local observer (same spatial position as the event)

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9
Q

how to measure the length of a moving object

A

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

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10
Q

how to measure length of a moving train

A

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

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11
Q

measuring lengths of moving objects - this approach relies on what 3 things?

A
  1. specific procedure for synchronising clocks
  2. assumes two events happening same time and place are simultaneous
  3. assume moving clocks measure the passage of time accurately
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12
Q

clock hypothesis

A

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

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13
Q

concrete example of a good clock

A

atomic clock

depends on fundamental physics to mark out a timescale

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14
Q

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:

A
  1. they are aligned so that the (x,y,z) and (x’,y’,z’) axes are parallel.
  2. frame S’ is moving along x-axis with velocity v.
  3. zero of time coords coincide (t=t’=0) so origin of S’ frame is always at xs’=vt in frame S
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15
Q

galiliean transformation

A

relates two frame in standard config (wrong)

x’=x-vt
y’=y
z’=z
t’=t

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16
Q

differentiating galilean transformation

A

diff once:
v’x=vx-v
v’y=vy
v’z=vz

diff again:

a’=a

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17
Q

equations of motion in reference frames

A

replace unprimed terms with primed ones

exactly the same physics

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18
Q

what proved galilean transformation wrong

A

Maxwell’s equations

if gt true, physics different in moving frames

maxwell correct
RP correct
gt only approximate
special relativity needed

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19
Q

principle of relativity

A

all inertial frames are equivalent for the performance of all physical experiments

(includes chemistry,biology etc)

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20
Q

second axiom

A

speed of light is a universal constant

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21
Q

only case where two events are unambiguously simultaneous

A

if they take place at exactly the same points in space

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22
Q

light clock

A

idealised timekeeper

flash of light leaves a bulb, bounces off a mirror and returns (one tick)

2L=c delta t’

this gives lorentz factor

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23
Q

light clocks conclusion

A

both clocks are perfectly accurately measuring the passage of time; time is flowing differently for the two observers

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24
Q

lorentz factor graph

A

even at nearly 90% of c, gamma=2

beyond this exponential

infinite at v=c

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25
Q

natural units

A

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

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26
Q

which axis is vertical in Minkowski diagram

A

t

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27
Q

moving train of Minkkowki diagram

A

diagonal (moving through space with time)

front and back of train lines parallel as same v

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28
Q

Relationship between the two sets of coordinates in Minkowski diagram

A

x’=xcos theta+ y sin theta
y’=-x sin theta + y cos theta

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29
Q

distance r is same in both frames

A

invarient of the transformation

x^2+y^2 = r^2 = x’^2+y’^2

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30
Q

invarient interval, delta s^2

A

delta t^2 - delta x^2

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31
Q
A
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32
Q

Space like separated

A

Interval -ve

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33
Q

Time like separated

A

Interval +ve

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34
Q

Light like separated

A

Interval 0

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35
Q

Relativity of simultaneity

A

Clocks will remain synchronised only in their own frame

36
Q

momentum 4-vector

A

P=mU=m gamma(1,v)

37
Q

length on momentum 4-vector in the rest frame of the particle

A

m which points along the particle’s world-line (direction of particle’s movement)

38
Q

pair of incoming particles which collide and produce set of outgoing particles

A

P1+P2=P3+P4

m1γ(v1)+m2γ(v2)=m3γ(v3)+m4γ(v4)

39
Q

as v approaches 0, γ(v)

A

approaches 1

40
Q

low-speed limit of the spatial part of the vector P is

A

just mv

ie momentum conserved

41
Q

time component of the 4-momentum

A

P^0=γm

42
Q

low v limit of time component of the 4-momentum

A

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)

43
Q

o(v^4)

A

order of v^4

44
Q

what is p^0

A

relativistic energy of a particle with mass m and velocity v

E=P^0=γm ie E=γmc^2

45
Q

magnitude of 4-vector will be frame-invariant so

A

will express something fundamental about the vector, analogous to its length

46
Q

P.P=

A

m^2

47
Q

why we cannot define a 4-velocity vector for a photon by the same route as before

A

dR.dR is always zero and so proper time is also zero

48
Q

alternative route for 4-velocity vector for a photon

A

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

49
Q

P^1_2 notation

A

component 1(x-component) of the vector P2

superscripts denote different momentum components

subscripts denote different particle momenta

50
Q

conserved

A

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

51
Q

covariant

A

refers to an equation which does not change its form when it is transformed form one frame to another

52
Q

invariant

A

refers to a quantity which is not changed by some transformation

this necessarily refers to more than one reference frame

53
Q

constant

A

refers to a quantity which does not change in time

54
Q

what can be considered conserved, invariant and constant

A

speed of light

55
Q

centre of momentum frame

A

frame in which the total spatial momentum is zero so that incoming particles have equal and opposite spatial momenta

56
Q

centre of mass energy

A

energy available for particle production in the centre of momentum frame

57
Q

in general relativity it is not mass that is the source of gravitation but

A

4-momentum

58
Q

compton scattering

A

an incoming photon strikes a stationary electron and both recoil

59
Q

in the context of GR, an inertial frame is

A

a frame in which SR applies and thus the frame in which the laws of nature take their corresponding simple form

60
Q

the equivalence principle

A

uniform gravitational fields are equivalent to frames that accelerate uniformly relative to inertial frames

61
Q

equivalence principle version 2

A

all local, freely falling non-rotating laboratories are fully equivalent for the performance of physical experiments

62
Q

the non-free-fall observer will measure light path of shining torch

A

as being curved in the gravitational field

ie even massless light is affected by gravity

63
Q

tidal effects

A

2 objects will fall towards centre of the Earth

64
Q

curvature of space is 1/R and as R gets larger

A

surface becomes locally more like flat euclidean space (curvature tends to zero)

65
Q

hyperbolic paraboloid

A

surface with constant negative curvature

internal angles of triangles add up to less than pi, circumference >piD

66
Q

metric of Minkowski space

A

ds^2=dt^2-dx^2-dy^2-dz^2

67
Q

geodasics

A

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

68
Q

equivalence principle, version 3

A

any physical law that can be expressed geometrically in SR has exactly the same form in a locally inertial frame of a curved spacetime

69
Q

spacetime tells matter…

A

how to move

70
Q

tensor

A

more general geometrical object to describe energy-momentum of an extended object

‘vectors squared’

71
Q

einstein tensor, G

A

specific tensor constructed in terms of second derivatives of the metric and therefore characterises curvature of spacetime that metric describes

72
Q

einstein’s equation

A

equation for gravity in which the constant k plays same broad role as G in Newton’s theory

G=-kT

73
Q

einstein’s equation represents a

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

74
Q

matter tells spacetime

A

how to curve

75
Q

if gravitating mass is small such as something around a planet this creates

A

the ‘weak-field’ approximation to the solution

76
Q

at speeds much less than that of light, near a ‘small mass’, the spacetime

A

is approximately that of empty space (approximately Minkowski space)

77
Q

a gravitational time dilation

A

a clock at a low altitude observed by an observer at a higher one is measured as slower than equivalent clock beside the observer

78
Q

what are the geodesics of minkowski spacetime

A

the r(t) that the geodasics traces out is the same as the motion of a test particle in a gravitational field

79
Q

Schwarzschild solution

A

metric in terms of the coordinates (t,r,theta, X) where theta and chi are the usual polar coords

80
Q

Schwarzschild radius

A

R=2GM

80
Q

asymptotically flat

A

at large r it reduces to the Minkowski metric

81
Q

why coefficient dr^2 is divergent (appears to be a singularity)

A

coordinate system goes wrong here

82
Q

the schwarzschild radius is the

A

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

83
Q

schwarzschild solution includes some geodesics which spiral around the t-axis as

A

near ellipses

84
Q

to find a metric for the universe as a whole, we start with the observation that

A

on the largest scales the universe appears homogeneous and isotropic (same at all spatial points and spatial directions)

this is the copernican principle

85
Q

cosmological constant

A

scaling factor Λ

86
Q
A

Astronomy 2 (4 decks)

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