4. Schwarzchild Solution Flashcards
Describe the Schwarzchild solution
A solution for the vacuum region of space time outside a single, central massive body
What are the important properties of the Schwarzchild solution?
It is exact and spherically symetric
How is the metric in the Schwarzchild solution different to the Minkowski metric?
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)
How are the functions, f(r) and g(r) found?
By inserting into the Einstein equations and using the Christoffel symbols, Ricci tensor and scalar
What is the mass parameter?
The only free variable in the Schwarzchild metric
- Dimensions of Length
- m = GM/c^2
What is the mass parameter of the sun?
m = 1.48km
What is the Schwarzchild radius, and what is it equal to?
- Coordinate singularity at r=2m
- This is not an actual singularity in space time
- It is the event horizon of the Schwarzchild black hole
When is there a physical singularity in the Schwarzchild metric?
At r = 0
- Black hole
State the values of the coordinate and physical singularities of the Schwarzchild metric
Coord: r=2m
Physical: r=0
What is the event horizon?
Where the metric is singular at r=2m
Describe what someone on board a space ship falling towards a black hole would experience time wise
They would arrive in finite time
- No divergence of space-time
Describe what someone observing a space ship falling towards a black hole would experience time wise
The time measured is logarithmically divergent as r->2m
- You never see the space craft arrive
How would light signals being sent from the space craft to the observer travel through space time?
Along radial null geodesics
Describe what is meant by the event horizon “concealing” a region of space time
- You can travel to it in a finite time
- If you stay behind and observe, it is forever hidden
What are the Eddington-Finkelstein coordinates?
Where ct is replaced with u
- u is the constant trajectory along radial null geodesics
What is the key difference between using the Schwarzchild metric and the Eddington-Finkelstein metric?
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
What does it mean if space time is static?
It is invariant under time reversal or t -> -t
What is a white hole?
The region complementary to a time reversed version of a black hole
How are the Kruskal coordinates first introduced?
Using both the Edd-Fin definitions for the ingoing (u) and outgoing (v) coordinates and putting them into the metric
What problem arises when using the Kruskal coordinates initially when plugging u and v into the metric, and how is it resolved?
The coordinate singularity of r=2m has now returned
- Transform it out by choosing U = exp(u/4m) and V = -exp(-v/4m)
Describe the two important results of the final version of the metric using the Kruskal coordinates
- 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
Describe the space time diagram using the Kruskal coordinates
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
Describe Schwarzchild space time
The behaviour of a small body in a spherically symmetric gravitational far-field of a larger mass
Describe gravitational lensing briefly
The observation of light travelling along null geodesics and passing near a massive object
Describe gravitational lensing briefly
The observation of light travelling along null geodesics and passing near a massive object
What is the value of Kappa when looking at gravitational lensing, and why?
Kappa = 0
- Light travelling along null geodesics
What does b represent in the deflection of light?
The impact parameter
- Distance of closest approach for the light
What is it called when the value of b is minimised, and why might it be useful?
- Min b is called the grazing incidence (radius of body)
- Total deflection = 4m/b and minimising b maximises deflection
What do the two values of the viewing angle, theta for gravitational lensing represent?
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)
Explain the necessary conditions for a “full Einstein ring”
The source must be on the same plane as the observer (x = 0)
- Light travels around all geodesics around the lensing object
Describe gravitational redshift
The difference perceived in frequencies from an area of strong and weak gravitational fields
What is meant by inferior and superior conjunction of a planet?
Whether the planet is at a point closest to ourselves in orbit, or furthest away
What is meant by inferior and superior conjunction of a planet?
Whether the planet is at a point closest to ourselves in orbit, or furthest away
What is meant by inferior and superior conjunction of a planet?
Whether the planet is at a point closest to ourselves in orbit, or furthest away
Describe briefly the Shapiro delay experiment
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
