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Aa General Relativity > 4. Schwarzchild Solution > Flashcards

4. Schwarzchild Solution Flashcards

1
Q

Describe the Schwarzchild solution

A

A solution for the vacuum region of space time outside a single, central massive body

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2
Q

What are the important properties of the Schwarzchild solution?

A

It is exact and spherically symetric

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3
Q

How is the metric in the Schwarzchild solution different to the Minkowski metric?

A

There are two additional unknown functions, f(r) and g(r) which depend on the radial coordinate
- Metric is stationary (t -> t + const) and static (t _. -t time reversal)

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4
Q

How are the functions, f(r) and g(r) found?

A

By inserting into the Einstein equations and using the Christoffel symbols, Ricci tensor and scalar

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5
Q

What is the mass parameter?

A

The only free variable in the Schwarzchild metric
- Dimensions of Length
- m = GM/c^2

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6
Q

What is the mass parameter of the sun?

A

m = 1.48km

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7
Q

What is the Schwarzchild radius, and what is it equal to?

A
  • Coordinate singularity at r=2m
  • This is not an actual singularity in space time
  • It is the event horizon of the Schwarzchild black hole
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8
Q

When is there a physical singularity in the Schwarzchild metric?

A

At r = 0
- Black hole

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9
Q

State the values of the coordinate and physical singularities of the Schwarzchild metric

A

Coord: r=2m
Physical: r=0

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10
Q

What is the event horizon?

A

Where the metric is singular at r=2m

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11
Q

Describe what someone on board a space ship falling towards a black hole would experience time wise

A

They would arrive in finite time
- No divergence of space-time

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12
Q

Describe what someone observing a space ship falling towards a black hole would experience time wise

A

The time measured is logarithmically divergent as r->2m
- You never see the space craft arrive

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13
Q

How would light signals being sent from the space craft to the observer travel through space time?

A

Along radial null geodesics

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14
Q

Describe what is meant by the event horizon “concealing” a region of space time

A
  • You can travel to it in a finite time
  • If you stay behind and observe, it is forever hidden
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15
Q

What are the Eddington-Finkelstein coordinates?

A

Where ct is replaced with u
- u is the constant trajectory along radial null geodesics

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16
Q

What is the key difference between using the Schwarzchild metric and the Eddington-Finkelstein metric?

A

The Edd-Fin metric doesn’t diverge at the Schwarz event horizon of r=2m
- Event horizon is a null surface
- Can calculate the area of a black hole

17
Q

What does it mean if space time is static?

A

It is invariant under time reversal or t -> -t

18
Q

What is a white hole?

A

The region complementary to a time reversed version of a black hole

19
Q

How are the Kruskal coordinates first introduced?

A

Using both the Edd-Fin definitions for the ingoing (u) and outgoing (v) coordinates and putting them into the metric

20
Q

What problem arises when using the Kruskal coordinates initially when plugging u and v into the metric, and how is it resolved?

A

The coordinate singularity of r=2m has now returned
- Transform it out by choosing U = exp(u/4m) and V = -exp(-v/4m)

21
Q

Describe the two important results of the final version of the metric using the Kruskal coordinates

A
  • There is still a singularity at r=0 (BH) which is important as it should not have been removed form a coordinate transform
  • Metric is also no longer singular at r=2m
22
Q

Describe the space time diagram using the Kruskal coordinates

A

Large central light cone (r=2m) with cT = y axis, X = x axis
- Top is a BH (r=0), bottom is a WH (r=0)
- Central point is the spherical portal (BH area 16 pi m^2) Einstein Rosen bridge
- Region I is on the right and is asymptotically flat described by (ct,r)
- Region II is the top, Region III is the bottom, region IV is the left
- Cannot travel from I to IV (space like). Trajectories just in one region are time like

23
Q

Describe Schwarzchild space time

A

The behaviour of a small body in a spherically symmetric gravitational far-field of a larger mass

24
Q

Describe gravitational lensing briefly

A

The observation of light travelling along null geodesics and passing near a massive object

24
Q

Describe gravitational lensing briefly

A

The observation of light travelling along null geodesics and passing near a massive object

25
Q

What is the value of Kappa when looking at gravitational lensing, and why?

A

Kappa = 0
- Light travelling along null geodesics

26
Q

What does b represent in the deflection of light?

A

The impact parameter
- Distance of closest approach for the light

27
Q

What is it called when the value of b is minimised, and why might it be useful?

A
  • Min b is called the grazing incidence (radius of body)
  • Total deflection = 4m/b and minimising b maximises deflection
28
Q

What do the two values of the viewing angle, theta for gravitational lensing represent?

A

Light that travels above and below the object which is lensing the light
Above lens: Minimum geodesic (image above source)
Below lens: Maximum geodesic (image below source)

29
Q

Explain the necessary conditions for a “full Einstein ring”

A

The source must be on the same plane as the observer (x = 0)
- Light travels around all geodesics around the lensing object

30
Q

Describe gravitational redshift

A

The difference perceived in frequencies from an area of strong and weak gravitational fields

31
Q

What is meant by inferior and superior conjunction of a planet?

A

Whether the planet is at a point closest to ourselves in orbit, or furthest away

31
Q

What is meant by inferior and superior conjunction of a planet?

A

Whether the planet is at a point closest to ourselves in orbit, or furthest away

31
Q

What is meant by inferior and superior conjunction of a planet?

A

Whether the planet is at a point closest to ourselves in orbit, or furthest away

32
Q

Describe briefly the Shapiro delay experiment

A

Experimenting with signals sent between Earth and Venus at different conjunctions
- E.g. at superior conjunction, the signal has to pass from Venus past the Sun’s gravitational field

Aa General Relativity (7 decks)

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