Using the Schwarzschild coordinates (t, r, 8, 4), the line element is

(1)

by the local stationary Schwarzschild observer along the way at radius r, satisfies

w. (r) (1-2M)

observer sees that locally there is a static, uniform gravitational field with acceleration magnitude

g. The observer sees that the photon falls in this gravitational field, and its observed frequency

turn Eq. (3) into a differential equation for w. (r). Solve the differential equation and reproduce

the result of Part (a).

Question 1

PART A

We want to show that the observed frequency w(r) as measured by a local stationary Schwarzschild observer along the photon's trajectory satisfies the equation:

w(r)√(1 - 2M/r) = const

p² = 0

Using the line element for the Schwarzschild metric in the given coordinates, we can calculate the contravariant components of the photon's four-momentum:

g^(tt)(E/c) - g^(rr)(pᵣ) = const

Substituting the expressions for the contravariant components of the metric, we get:

hf/c = const (1 - 2M/r) + (pᵣ)(1 - 2M/r)

Simplifying further:

w(r)√(1 - 2M/r) = const

Therefore, the observed frequency of a free photon falling radially toward the event horizon of a Schwarzschild black hole, as measured by the local stationary Schwarzschild observer, is constant multiplied by the term √(1 - 2M

PART B

Using the Schwarzschild metric in the given coordinates, we can calculate the non-zero components of the Christoffel symbols:

Γ^r_tt = (1/2)g^rr (d/dt)(g_tt) = (1/2)g^rr (d/dt)(-(1 - 2M/r)) = M/r²

Next, we can use the relation dw = w·g·ds to derive a differential equation for w(r). The vertical proper distance ds can be written as ds = √(g_rr)dr, where g_rr is the rr-component of the metric tensor.

Substituting these values and integrating, we get:

ln|w| = -M∫(√r/(r - 2M))·dr = -M∫(du/√(2M)) = -√(2M)·ln(u)

Substituting back u = √(r/(r - 2M)), we have:

|w| = (√(r/(r - 2M)))^(-√(2M))

Taking the absolute value of both sides and simplifying:

PART C

To determine the thrust required for the space station to suspend the cable in stationarity, we can consider the tension in the cable at different points along its length. Let's analyze a small segment of the cable at a specific radius r, as seen by a local stationary Schwarzschild observer.

According to Newtonian mechanics, the tension T in the cable changes as the cable is lowered in a gravitational field. The change in tension can be related to the local gravitational acceleration g and the proper length ds of the infinitesimal cable segment by the equation:

g·dm - T = 0

Substituting dm = A·ds and rearranging the equation, we have:

Substituting this expression into the force balance equation:

-(M/r²) √(r/(r - 2M))·A·ds - T = 0

The integral on the left-hand side represents the total thrust provided by the space station. To evaluate the integral on the right-hand side, we need to express ds in terms of the Schwarzschild coordinates.

The proper length ds can be related to the line element ds² as:

Taking the square root of both sides:

ds = √(1 / (1 - 2M/r))·dr

Combining the square roots:

∫[R₁→R₂] T = -A∫[R₁→R₂] (M/r²)·√(r² / (r - 2M))·dr

Next, we can split the integral into two parts by using partial fraction decomposition. Let's write the expression as:

∫[R₁→R₂] T = -A∫[R₁→R₂] (M/r²)·(√r / (r - 2M))·(√(r - 2M) + √r)·dr

To evaluate this integral, we can express it in terms of the photon's observed frequency w(r) as derived in Part (a):

∫[R₁→R₂] (M/r²)·√r·dr = ∫[R₁→R₂] (M/r²)·√(r/(1 - 2M))·√(1 - 2M)·dr = ∫[R₁→R₂] (M/r²)·w(r)·√(1 - 2M)·dr

Combining the integrals:

∫[R₁→R₂] T = -A∫[R₁→R₂] (M/r²)·w(r)·(√(1 - 2M) + 1)·dr

Simplifying further:

∫[R₁→R₂] T = -A(M/√(1 - 2M))·[w(R₂)·(√(1 - 2M) + 1)]·[R₂] + A(M/√(1 - 2M))·[w(R₁)·(√(1 - 2M) + 1)]·[R₁]

Hence, as the lower end of the cable approaches the horizon, the required thrust goes to zero. This indicates that the space station does not need to provide any thrust to suspend the cable when it reaches the horizon. The gravitational force alone can support the cable's weight in stationarity.