Schwarzschild Solution & Black Holes

Question 1

What is the Schwarzschild metric?

Answer 1

ds² = -(1 - 2GM/r)dt² + (1 - 2GM/r)⁻¹dr² + r²dΩ².

Question 2

What symmetry does the Schwarzschild solution assume?

Answer 2

Spherical symmetry and vacuum.

Question 3

What is the Schwarzschild radius?

Answer 3

r_s = 2GM/c².

Question 4

What happens at r = r_s?

Answer 4

The event horizon of a black hole — a coordinate singularity.

Question 5

Is the singularity at r = 2GM physical?

Answer 5

No, it’s a coordinate artifact; can be removed with new coordinates.

Question 6

Where is the true singularity in Schwarzschild?

Answer 6

At r = 0, where curvature invariants diverge.

Question 7

What coordinates remove the horizon singularity?

Answer 7

Eddington-Finkelstein or Kruskal-Szekeres coordinates.

Question 8

What is a black hole?

Answer 8

A region of spacetime from which nothing can escape.

Question 9

What is an event horizon?

Answer 9

A boundary beyond which events cannot affect an outside observer.

Question 10

What is the proper time for infalling observer at horizon?

Answer 10

Finite — they cross in finite proper time.

Question 11

What happens to dt as r → r_s in Schwarzschild?

Answer 11

dt → ∞ — coordinate time diverges for an infalling object.

Question 12

What is dΩ² in the metric?

Answer 12

dΩ² = dθ² + sin²θ dφ² — the angular part of the metric.

Question 13

What is the escape velocity at r_s?

Answer 13

Equal to the speed of light, c.

Question 14

What does g_{tt} = 0 mean?

Answer 14

Time slows to zero from the perspective of a distant observer.

Question 15

What is the gravitational redshift near a black hole?

Answer 15

Light emitted near r_s is redshifted infinitely at infinity.

Question 16

What defines the Schwarzschild black hole mass?

Answer 16

The M parameter in the metric — related to the ADM mass.

Question 17

Is the Schwarzschild black hole rotating?

Answer 17

No, it is static and spherically symmetric.

Question 18

Can light escape from r < r_s?

Answer 18

No, not even light can escape the event horizon.

Question 19

What is a coordinate singularity?

Answer 19

A singularity removable by a change of coordinates (e.g. r = 2GM).

Question 20

What is a curvature singularity?

Answer 20

A true physical singularity where invariants like R_{μνρσ}R^{μνρσ} diverge.

Question 21

What is the effective potential for Schwarzschild geodesics?

Answer 21

Derived from the conserved energy and angular momentum.

Question 22

What is the ISCO?

Answer 22

Innermost Stable Circular Orbit — r = 6GM for Schwarzschild black holes.

Question 23

What is the photon sphere?

Answer 23

At r = 3GM, photons can orbit the black hole in unstable circular orbits.

Question 24

What does the Schwarzschild metric reduce to at large r?

Answer 24

Flat Minkowski space: ds² ≈ -dt² + dr² + r²dΩ².

Question 25

What is the gravitational time dilation?

Answer 25

Clocks near massive objects run slower relative to distant clocks.

Question 26

How is proper time computed in Schwarzschild?

Answer 26

Using the Schwarzschild metric with given worldline.

Question 27

What is Kruskal extension?

Answer 27

A maximal analytic extension of Schwarzschild spacetime across horizon.

Question 28

What is the Penrose diagram?

Answer 28

A conformal diagram representing causal structure of spacetime.

Question 29

What is the maximal extension of Schwarzschild?

Answer 29

A spacetime with two asymptotically flat regions and a black/white hole.

Question 30

What are the Schwarzschild coordinates?

Answer 30

t, r, θ, φ — standard static coordinates for black holes.

Question 31

Why do Schwarzschild coordinates break down at r_s?

Answer 31

Because g_{tt} and g_{rr} become singular.

Question 32

How is Hawking radiation related to Schwarzschild?

Answer 32

Quantum effect near horizon causes black hole to emit radiation.

Question 33

What does the Schwarzschild solution solve?

Answer 33

Einstein's field equations in vacuum, R_{μν} = 0.

Question 34

What is the causal structure inside r_s?

Answer 34

All future-directed paths point toward r = 0.

Question 35

Can Schwarzschild describe a star?

Answer 35

Only outside the star (r > R); inside requires different interior solution.

Question 36

What is the Buchdahl limit?

Answer 36

The minimum radius for a stable star before collapse: R > (9/8)r_s.

Question 37

Is the Schwarzschild metric static?

Answer 37

Yes, it has a timelike Killing vector field outside the horizon.

Question 38

What is the physical meaning of mass M?

Answer 38

The source of gravitational field, measurable from infinity.

Question 39

What is the geodesic motion in Schwarzschild?

Answer 39

Freely falling objects follow curved paths governed by the metric.

Question 40

What distinguishes a black hole from a star?

Answer 40

A black hole has an event horizon and central singularity.