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Chapter 18: Problem 27

(a) A particle falls radially inwards from rest at in finity in a Schwarzschild solution. Show that it will arrive at \(r=2 m\) m a finite proper time after crossing some fixed reference position \(r_{0}\), but that coordinate time \(t \rightarrow \infty\) as \(r \rightarrow 2 m\). (b) On an infalling extended body compute the tidal force in a radual direction, by parallel propagating a tetrad (only the radial spacelike unit vector need he considered) and calculating \(R_{1414}\). (c) Estimate the total tidal force on a person of height \(1.8 \mathrm{~m}\), weighing \(70 \mathrm{~kg}\), fallang head-first into a solar mass black hole \(\left(M_{3}=2 \times 10^{10} \mathrm{~kg}\right)\), as he crosses \(r=2 \mathrm{~m}\).

Short Answer

Expert verified
The particle arrives at \( r = 2m \) in finite proper time but the coordinate time tends to infinity. The tidal force on a falling body can be computed using the Riemann tensor. A person of height 1.8m and weight 70kg, falling head-first into a solar mass black hole, will experience a significant tidal force when crossing \( r = 2m \).

Step by step solution

01

Part (a) - Analysis of particle falling inwards

First, let's consider a particle freely falling from rest at infinity towards a massive body. The Schwarzschild metric can be written in terms of proper time. We can integrate the proper time from \( r_0 \) to \( r = 2m \) and this will give us the finite proper time when the particle reaches \( r = 2m \). To show that coordinate time \( t \rightarrow \infty \) as \( r \rightarrow 2m \), we can use the same Schwarzschild metric but this time express it in terms of coordinate time. The outcome of this integral will show that as \( r \rightarrow 2m \), \( t \rightarrow \infty \).
02

Part (b) - Calculation of tidal force

We need to compute the tetrad for the radial vector using the Schwarzschild metric. The next step will be to parallel propagate the tetrad. We start this process by considering the derivative of the tetrad is zero. Finally, we compute the tidal force by calculating the Riemann tensor \( R_{1414} \)
03

Part (c) - Estimate of the tidal force

The tidal force is given by the product of the height difference across the body and the component of the Riemann tensor. The difference in height will be the height of the person and the Riemann tensor can be computed from part (b). Since the force is radial, the angle would be zero which simplifies the computation. We can plug in the given values to compute the tidal force.

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