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PHYS55390M General Relativity > Schwarzchild Metric > Flashcards

Schwarzchild Metric Flashcards

1
Q

Schwarzchild Metric

Description

A

-spherical symmetry - angular part of the metric:
r²(dθ² + sin²θ dφ²)
-not homogeneous - gravitational field varies with position so dt ≠ dτ
-use static spacetimes:
1) no time derivatives of gμν
2) the geometry should be the same if t -> -t

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2
Q

Schwarzchild Metric

Metric

A

ds² = goo c² dt² + grr dr² + r²(dθ² + sin²θ dφ²)
-with goo and grr functions of r only
-often written as:
ds² = - e^2α c²dt² + e^2β dr² + r²(dθ² + sin²θ dφ²)
-where α and β are functions of r only

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3
Q

Schwarzchild Metric

External

A 
-for the space between stars
=>
T^αβ = 0 and T = 0
-hence R00 = R11 = R22 = R33 = 0
-can use these conditions to find α and β
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4
Q

The Schwarzchild Radius

Definition

A

Rs = 2GM/c²

-Rs varies with the mass you have

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5
Q

The Schwarzchild Radius

Blackhole

A

-for a blackhole all of the mass, M, needs to be within the Schwarzchild radius

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6
Q

When does the Schwarzchild metric apply?

A
  • around a static star with spherical symmetry
  • for a blackhole
  • in empty space
  • with time independent (‘stationary’) motion, around a rotating star
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7
Q

Schwarzchild Metric

r > > Rs

A

-metric goes towards the weak field Schwarzchild metric

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8
Q

Schwarzchild Metric

Rs/r -> 0

A
  • recover the flat space metric of special relativity
  • spacetime far away from a star is described as being ‘asymptotically flat’
  • this is also proper time: dτ -> dt(r=0)
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9
Q

Schwarzchild Metric

Near Rs

A

-clocks run slower in a gravitatoinal potential:

dt > dτ

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10
Q

Clocks Near Earth

A
  • the clock at the pole experiences only the gravitational effect
  • the clock on the equator sees the same gravitational effect but also experiences time dilation due to its motion
  • the clock in orbit (geostationary) sees less of a gravitational effect because its further away but a larger effect due to its motion (satellite has to move faster than surface to maintain same relative position because its at a greater radius)
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11
Q

Constants of Motion

A

-from the Euler Lagrange Equations:
r²φ’ = const. = h
-conservation of angular momentum, where h is the angular momentum per unit mass
t² (1 - Rs/r) = const. = K
-conservation of energy where K is the energy per unit rest mass

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12
Q

Kepler’s Laws

A

-found from the Geodesic equation in r with dr=0 (circular motion) , θ’=0 and θ=π/2
=>
(dφ/dt)² = GM/r³
-this is true for circular obits even when r is not greater than Rs

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13
Q

Full Orbit Equation

Derivation

A

-use:
U^μ Uμ = -c² = gαβ U^α U^β = gαβ x^α’ x^β’
-with θ = π/2 and λ=τ for massive particles
-sub in t’ and φ’ from the conservation equations
-rearrange and multiply by 1/2

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14
Q

Full Orbit Equation

Word Equation

A

kinetic term + potential term = total energy

-a scaled energy equation per unit mass

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15
Q

Full Orbit Equation

Potential

A

V(r) = - 1/2 Rs/r c² + 1/2 h²/r² - 1/2 Rs h²/r³

= - GM/r + 1/2 h²/r² - GM/c² h²/r³

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16
Q

Full Orbit Equation

Newtonian Comparison

A

Vn(r) = 1/2 h²/r² - GM/r
= centrifugal barrier term + real potential
-so GR differs by the addition of a 1/r³ term

17
Q

Schwarzchild Metric

Photons

A
  • still have the conservation equations

- but now derivatives are with respect to λ not τ

18
Q

Full Orbit Equation

Potential for Photons

A

V = 1/2 h²/r² - GM/c² h²/r³

-no Newtonian 1/r term for photons

19
Q

Characteristics of Orbits

Circular Orbits

A

-circular orbit:
r’=0
AND
V’(r) = 0

20
Q

Characteristics of Orbits

Solutions for Massive Particles

A
  • case h < √3 Rs c - no solution so there are no circular orbits, at low angular momentum particles MUST fall towards the centre
  • case h = √3 Rs c - minimum radius of circular orbit, inflection point so not stable
  • case h > √3 Rs c - looking at the limit h > > √3 Rs c gives two solutions
    • a maximum of V so unstable and a minimum which is potentially stable
  • -it goes to increasingly large radii as h increases
21
Q

Plotting Orbits

A
  • in the units used, 2V/c² is dimensionless

- parameterise h in terms of Rs c, i.e. h = m Rs c with m a dimensionless number

22
Q

Plotting Orbits

GR

A

massive particles

  • potential barrier
  • from unstable orbit:
  • -could be knocked outwards and drift away
  • -of fall in
  • there is a stable orbit radius

photons
-only unstable orbits for photons

23
Q

Plotting Orbits

Newtonian

A

massive particles

  • infinite potential barrier, if you have any angular momentum at all you can never hit the central source
  • all orbits are stable

photons

  • also infinite potential barrier but no orbits
  • photons can be deflected but will always continue on past
24
Q

Radial Trajectories - Photons

Equation

A

-for radial trajectory photons; ds²=0 and dθ=dφ=0
=>
dr/dt = ±c [1 - Rs/r]
-with + for outward and - for inward photons

25
Q

Radial Trajectories - Photons

Observers / Measuring Velocity

A

-r is not a proper length unless measured at r=∞, the same is true for t
-an observer experiences a time dτ and sees proper lengths dl so a measurable velocity for an observer is dl/dτ = c
-i.e. to a local observer light appears to still travel at c
-for a observer at r=∞ they see light as if its moving in a medium of refractive index:
1 / √[1 - Rs/r]

26
Q

Radial Trajectories - Massive Particles

r < Rs

A

-massive particles are timelike: ds² < 0
-if r < Rs and dr²=0 then ds² > 0 -> contradiction
-so we must have dr² ≠ 0
=>
-no stationary observers for r < Rs - everything falls towards the centre

-OR the only way to have non-stationary observers within the event horizon is for them to be moving faster than c which is not possible

27
Q

Radial Trajectories - Massive Particles

Observers / Measuring Velocity

A

-find dr/dτ by subbing in the constant of motion equation with K
-a fixed observer at radius r instead sees:
v = dl/dτ = …
-as r -> Rs, v -> c as it should at the event horizon
-if the observer is at ∞ instead they measure the rate of change of dl with respect to their clock dτ(r=∞)=dt which gives v -> 0 as r -> Rs

28
Q

Gravitational Redshift

Overview

A
  • imagine firing off a signal from a stationary emitter to a statoinary observer then doing the same again after a coordinate time difference Δt
  • the observer receives these separated by the same coordinate time difference Δt, but they will have different clock times
29
Q

Gravitational Redshift

Equation

A

νo/νe = Δτe/Δτo = √[1 - Rs/re] / √[1 - Rs/ro]

  • if ro > re then νo < νe
  • ir re -> Rs then νo -> 0
  • can’t see anything ar r=Rs if we are at larger r
30
Q

Gravitational Redshift

φ

A

-if ro, re > > Rs:
νo/νe ~ 1 - (Δφ/c²)
-if Δφ = φ(ro) - φ(re)

31
Q

The Event Horizon

Photons

A
  • consider a radially orbitting photon
  • light cones at r > > Rs have gradient ±1 i.e. 45 degree cones
  • they close up as r -> Rs (the gradient -> ∞)
  • for r < Rs the sign of the gradient flips and as r -> 0 the gradient -> 0 and the cones are completely closed and looking at the singularity at r=0
32
Q

The Event Horizon

Massive Particles

A
  • starting at rest near r=∞ (so k²-1=0 since there is no kinetic or potential energy)
  • the proper time that elapses in going from r1 -> r2 is finite regardless of where r2 is, even if r2=0
  • a particle falling in from any finite distance, BY ITS OWN CLOCK will experience a finite time to reach the central singularity and no boundary exists at r=Rs
  • an observer from outside sees the particle slowing down (as it approaches Rs) and never crossing the event horizon

PHYS55390M General Relativity (7 decks)

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