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What does the term Euclidian describe?
The metric of flat space.
What are the two ways by which curvature can be understood?
This can be understood geometrically or by way of metric equations.
What is an example of how the geometry of a sphere is not the same as that of a flat surface?
The sum of the angles of a triangle on the surface of the sphere is larger than 180 degrees.
What kind of geometry does a surface with k = 0 have?
Flat geometry, C = 2π a, k = 0
What kind of geometry does a surface with curvature k < 0 have?
Hyperbolic geometry on a saddle shaped surface, C > 2π a so there is negative curvature as k < 0.
What kind of geometry does a surface with curvature k > 0 have?
Spherical Geometry on a sphere or dome-shaped surface, C <2π a so there is a positive curvature as k > 0.
Explain what a metric space is and what the metric is.
A metric space is one where a distance in space is well defined.
The metric is the object which describes how distance is related to coordinates (e.g. x, y or r, φ)
Why for k>0, does the equation shown indicate a 'closed space'?
A is periodic so increases with proper distance s until a certain point and then decreases to 0 so there is a limit to the size A can be.
Why for k<0, does the equation shown indicate an 'open space'?
A will expand for all s, hence the space is 'open'
2-sphere
The curvature is an intrinsic property that is implicit to us but becomes explicit on the curved surface of a sphere because the same curvature is also a property of that of a 2D surface. The analogue of the surface of a sphere is called a 2-sphere.
What are fundamental observers?
A set of observers at different locations who are at rest w.r.t the matter in their vicinity, they form local inertial frames.
What is the full metric equation for an isotropic and homogeneous universe given by?
The Robertson-Walker metric (RW or FLRW or FRW metric)
Considering two events separated by dt and dx, what does it mean if ds^2 > 0?
This is a 'time like' interval when ds^2 = c^2 dτ^2 where τ is proper time. A casual relation can exist between between events.
Considering two events separated by dt and dx, what does it mean if ds^2 < 0?
This is a 'space like' interval when -ds^2 = dσ^2 where σ is proper distance. No casual relation can exist, no object can follow such a path.
Considering two events separated by dt and dx, what does it mean if ds^2 = 0?
This is a 'light like' interval for photons travelling in a vacuum.
What is the Schwarzschild metric?
The solution for the gravitational field outside a spherically symmetric, non-rotating body of mass M.
What happens to the Schwarzschild metric when M --> 0?
At this point the Schwarzschild metric becomes the same as the FRW metric.
What is the Schwarzschild metric used for?
This is used to calculate the precession of the perihelion of Mercury due to General Relativity.
Why does the full GR result give twice the light deflection predicted by EP or Newtonian theory?
There is equal contribution from g_11 (time-time) and g_22 (curved space) terms.
What is gravitational redshift caused by in General Relativity?
It is caused by time dilation in general relativity.
What is different about the relationship between r_s and r in black holes?
For most masses, r is > > r_s, however in black holes all the mass is 'contained' within r_s
What is the radial coordinate called at r_s = r
This radial coordinate is called the event horizon, no information can escape when r is less than or equal to r_s.