The Schwarzschild solution: orbits, redshift and the horizon · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Clocks, orbits, and the edge of a black hole

Science · Astronomy & Astrophysics · ages 23-24
Name ______________________   Date ____________
  1. Compared with an isolated Newtonian point-mass ellipse, what does the Schwarzschild correction do to a weak-field bound orbit?

    • It makes the nearest point retreat each orbit
    • It advances the nearest point each orbit
    • It removes every change in the planet's direction
  2. Two static clocks each remain at fixed r, θ and φ outside rₛ in a Schwarzschild vacuum region. Their radii differ. During the same interval of t, which accumulates less proper time?

    • The clock at the smaller radius
    • The clock at the larger radius
    • Both clocks, regardless of their different radii
  3. A spherical star with its surface outside rₛ must have a black-hole horizon in its Schwarzschild vacuum exterior.

    Circle one:   True   False

  4. For the same spherical mass in the weak-deflection regime, a light ray's impact parameter b doubles. What happens to its deflection angle?

    • It becomes half as large
    • It becomes twice as large
    • It becomes four times as large
  5. Static clocks are at r_low=2rₛ and r_high=8rₛ. For the same interval of t, what is Δτ_low/Δτ_high?

    • √(7/4)
    • 4/7
    • √(4/7)
  6. Which statement correctly describes the Schwarzschild black-hole horizon?

    • It is a solid surface where every curvature measure diverges
    • It is a causal boundary with removable coordinate failure
    • It is only a label and has no effect on signal escape
  7. A report assigns all measured planetary precession to the Schwarzschild correction. What must it also check?

    • Newtonian perturbations
    • Whether every orbit has exactly the same radius
    • Whether changing coordinate names removes all precession
  8. A probe falls across an ideal Schwarzschild horizon. Does smooth geometry there guarantee that the probe survives?

    • Yes; smooth geometry removes all tidal forces
    • No; the horizon is a solid wall that stops it
    • No; finite tidal forces can still be large
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Answer key

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Clocks, orbits, and the edge of a black hole W1-mt_dNZpRDeZUE-s1

  1. It advances the nearest point each orbit · The relativistic correction produces a forward periapsis advance.
  2. The clock at the smaller radius · The factor √(1−rₛ/r) is smaller at the smaller radius.
  3. False · Its vacuum exterior begins outside the stellar surface, which lies outside rₛ.
  4. It becomes half as large · The approximation 2rₛ/b is inversely proportional to b.
  5. √(4/7) · The factors f are 1/2 and 7/8. Their proper-time ratio is √[(1/2)/(7/8)]=√(4/7).
  6. It is a causal boundary with removable coordinate failure · Regular coordinates cross the horizon, but signals inside cannot escape to distant observers.
  7. Newtonian perturbations · Other bodies and the source's shape can cause precession before the relativistic contribution is isolated.
  8. No; finite tidal forces can still be large · The horizon has no curvature divergence, but finite tidal forces can damage a probe.
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