Problem 27 - A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Question

(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 \(r=2 \mathrm{~m}\).

Step-by-Step Solution

Answer

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 1: 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 \( r \rightarrow 2m \), \( t \rightarrow \infty \).

Step 2: 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 \( R_{1414} \)

Step 3: 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.

Key Concepts

Tidal Forces

When an object approaches a massive body such as a black hole, tidal forces become a significant factor. These forces occur due to the gravitational gradient; different parts of the object experience different gravitational pulls. In a Schwarzschild solution scenario, objects falling into a black hole will be stretched and squeezed by tidal forces.
Tidal forces in the radial direction can be computed using components of the Riemann tensor. Specifically, for an infalling body, the radial tidal force can be measured by calculating the component of the Riemann tensor, denoted as \( R_{1414} \).

• The calculation involves parallel propagating the radial spacelike unit vector.
• This process helps to determine how vectors change as they move through the curved spacetime near a black hole.

Estimates of tidal forces on a human-sized object falling into a black hole can reveal the extreme conditions near such massive celestial bodies.

Riemann Tensor

In general relativity, the Riemann curvature tensor is a crucial mathematical object. It describes how the geometry of space is curved by gravity, providing deep insights into the structure of spacetime.
The tensor contains information about tidal forces and how matter moves through curved spacetime.
For a Schwarzschild metric, calculating specific components of the Riemann tensor, such as \( R_{1414} \), allows us to quantify tidal forces experienced at different radial distances.

• This component can be derived by considering how test particles are accelerated relative to each other due to spacetime curvature.
• In practical terms, the Riemann tensor helps predict tidal effects on objects moving within a gravitational field.

Understanding the Riemann tensor is essential for calculations involving complex motions in strong gravitational fields, like those near black holes.

Proper Time

Proper time is a concept from relativity used to measure time elapsed according to a clock that a moving observer carries. It's crucial for understanding how time is perceived under different gravitational conditions.
In a Schwarzschild solution, proper time can be integrated from a reference point to another to find how much time an observer experiences while falling through a gravitational field.

• For example, a particle falling into a black hole from infinity will take a finite proper time to reach the event horizon at \( r = 2m \).
• This proper time is what the observer experiences personally, irrespective of external coordinate measurements.

Proper time contrasts with coordinate time, showcasing the subjective experience of time for observers in strong gravitational fields.

Coordinate Time

Coordinate time is an important concept that differs from proper time, especially in the context of general relativity as it involves time measured by distant observers.
In a Schwarzschild solution, coordinate time becomes significant when evaluating how different observers measure the same event.

• As a particle approaches the event horizon of a black hole, coordinate time tends to infinity, even though the falling observer experiences only finite proper time.
• This means that from the perspective of an observer far from the gravitational field, the particle never appears to cross the event horizon.

Understanding the distinction between coordinate time and proper time aids in appreciating how relativity alters our perception of time depending on gravitational influences.