My last two posts were the first two chapters of my paper discussing Hawking radiation and black holes. This is a continuation of the same. One must read the first two chapters to understand this one properly. The links for the same are given below:

This material in this chapter borrows extensively from chapters 7 and 8 of [3]

Metric:

We use a system of four coordinates to describe the geometry of a system. This system is defined using a mathematical object called a metric and it relates the different dependent coordinates among each other and describes the geometry of the system. It is a matrix that shows how flat or curved a given system is, what singularities it has, etc.

The metric is written using the following syntax:

This metric is a position-dependent matrix given by:

Curved space-time:

Flat spacetime has multiple symmetries that can be used to simplify many of the metric equations. The choice of coordinates in curved spacetime is very important. As we shall see later in the black holes chapter the choice of coordinate system can determine whether a coordinate singularity forms at a certain point or not. This means that the metric function will tend to infinity at a certain point even if there is nothing actually happening there physically. When we change our coordinate system, the singularity disappears.

Local Inertial Frames:

Like we defined inertial frames in special relativity, there is a need to define these for general curved spacetime too.

It is given that every point has three spatial coordinates to define its position in space and one time coordinate to define its moment in time.

We can treat a very small region of curved spacetime to be flat. This is defined as a local inertial frame. It is described mathematically when the general metric is converted into a diagonal metric by multiplying and dividing with appropriate values and the first derivatives of this metric vanish.

Light Cones and World Lines in Curved space-time:

Light cones in curved space-time can be considered in the same way we have been considering them for flat space-time in special relativity.

The world lines of particles are timelike due to the universal speed limit placed on everything. They are expressed as functions of the distance between two points. The distance between 2 points in curved space-time is given by the following equation:

(The summation convention (gab), distance formula, etc have been discussed in the first 4 sections of Chapter 7 of [3])

This means that the light cones in curved space time behaves in ways similar to those of flat space time. However, all the light cones taken together do display some different properties.

These properties will be used later on in black hole space times. Example 7.4 of [3] explains an unrealistic situation of warp drive and how the light cones adjust themselves to avoid the breaking of fundamental laws.

Length, Area, Volume and Four Volume:

It is often needed to know the length, area, volume etc. of curves in curved, or even flat, spacetime. The equation for the distance has already been given above. In this case we will use a s metric such that the distance can be expressed as:

Look at the figure below given in the x1-x2 plane by taking x0 and x3 as a constant.

The area is given as dl1dl2 which is simplified as:

Similarly, the four volume is given as:

The minus sign is there to make the root real when applying the equation as, by convention, the time coordinate is given a negative sign.

Examples 7.5 and 7.6 in chapter 7 of [3] give good examples of how these results are applied in real situations.

Vectors in Curved Spacetime:

We cannot exactly define vectors in curved spacetime how we do in flat spacetime. We define vectors in curved spacetime locally. This means that it is in a very small region of spacetime. These are small vectors. We can create large vectors by adding, multiplying etc. these small vectors.

We have to abandon the idea of displacement and position vector since these are not local ideas with small distances.

Just like in flat spacetime vector algebra, we define basis vectors which express the vector as a linear combination:

The scalar product of vectors is given as:

We can pick whatever basis we want for this scalar product.

Orthonormal Bases:

In this we pick four mutually orthogonal basis vectors of unit length.

A scalar product of the basis vectors of A and B give a diagonal metric of the form diag (-1,1,1,1). The final scalar product of A and B is given as:

The momentum of the particle is given as:

As discussed in the special relativity chapter, where we talked about the four-momentum of an object, the energy of the particle is given by E = p0. Where p0 is the time basis. Therefore E = -p.uobs

Coordinate Bases:

We know that ua = dxa/dt.

To find the basis vectors we get that:

-1 = gab(dxa/dt dxb/dt)

The basis vectors are seen to be:

ea(x).eb(x) = gab

This means that the coordinate basis are not unit vectors and nor orthogonal.

For some more practice in using these orthonormal and coordinate bases, the reader can consult example 7.8, 7.9 and 7.10 of [3].

Let us now talk about the concept of geodesics.

Geodesics:

To study the concept of geodesics we will first introduce a related idea called: ‘test particles’

Test particles are bodies with relatively less mass (i.e., they don’t produce any significant spacetime curvature on their own) which move according the shape of space-time and respond to the curvature produced by other bodies.

A geodesic is defined as the shortest path between 2 points on any surface (curved or flat).

In case of flat planes, geodesics are merely straight lines.
On a sphere, a geodesic will be a curved line that joins two points. (Look at the diagram below)

Let us define another important term: proper time.

Proper time is essentially the time measure by a clock that moves along with the test particle along its world line.

The geodesic equations follow a variational principle that states that the world line of any test particle between two points which maximizes the proper time between them.

We will study how equations of motion change for curved spacetime. To define it we will use a general metric gαβ(x).

As an example, we will derive the geodesics in flat spacetime (straight lines) using the geodesic equations.

This is a discussion of example 8.1, chapter 8 of [3].

The metric in polar coordinates of a plane in terms of r and φ:

dS2 = dr2 + r22

The derivation of the equations of the geodesic involves lagrangian Mechanics which is beyond the scope of this paper. However, the final equations are given below:

Although, this is a complicated derivation, if the reader wants to try to derive this, it is available in example 8.1 of [3].

The general form of geodesic equations is given below:

The coefficient is called a Christoffel coefficient which consists of first derivatives of the metric.

From the general equations of flat space that were stated earlier we can see that the Christoffel coefficients are as follows:

(The reader can check example 8.3 of chapter 8 of [3] for more information)

A general equation for Christoffel coefficient can also be found which is given as:

This basically explains the Christoffel coefficient in terms of the metric and the partial derivatives of its coordinates.

Note: we will not be using these coefficients much in our discussion of black holes and Hawking radiation in this paper.

Killing Vectors:

One important aspect of geodesics is finding different symmetries in the system. These symmetries can be used to find conserved quantities like energy, momentum etc.

Conserved quantities are related to some symmetries found in the metric (it is independent of some coordinate). For example, to conserve energy there needs to be a symmetry in displacements in time, for linear momentum a symmetry in displacements in space is needed, and for angular momentum, symmetry in rotations is needed.

These symmetries are explained in terms of a vector called the killing vector, named after Wilhelm Killing. These vectors are along the independent coordinate. For example if a metric is independent of x1 then the killing vector has components (0,1,0,0).

Let us look at an example of killing vectors in flat space (8.6 of [3])

The metric is given as dS2 = dx2 + dy2 + dz2

Three killing vectors are seen for each translational symmetry, i.e., (1,0,0), (0,1,0), and (0,0,1).

The conserved quantity is given as:

Where ξ is the killing vector and u is the tangent vector. This will give the conserved quantity.

Example:

We will now derive the equation of a straight line for a flat plane using killing vectors. We will use polar coordinates.

It has been stated before that the metric for flat space is given by:

dS2 = dr2 + r22

Taking x1 = r, x2 = φ, tangent vector u = dxA/dS.

It is known that u. u = 1. Dividing both sides by dS2 we get:

The metric is independent of φ so the conserved quantity is:

(As the killing vector lies in this direction.)

Thus we get:

Which upon integration yields:

After simple rearranging of terms and taking cos on both sides gives:

Which is the familiar equation of a straight line.

(This derivation has been taken from example 8.7 in chapter 8 of [3])

Killing Vectors – Examples:

To get a better understanding of killing vectors, we can look at some applications of these:

First, let us look at something called Schwarzschild geometry. This is an important geometry and is heavily used in black hole physics as we will see later.

Schwarzschild Geometry:

We will now discuss the properties of a special type of curved spacetime.

The curved spacetime outside a spherical star is the simplest as it has the most symmetry.

The line element of this geometry is given by:

It is observed that the metric of Schwarzschild geometry has the following symmetries:

  1. This metric is independent of time:
    This means that there is a killing vector along the time axis of this metric. This vector, ξ is of the form (1, 0, 0, 0)
  2. It is spherically symmetric:
    This system has the symmetries of a three-dimensional sphere in flat spacetime. This essentially means that it is independent of the last φ angle in the four coordinates. This killing vector η is of the form (0, 0, 0, 1)

The conserved quantities are therefore given by ξ.u and η.u where u is the four velocity. These quantities are so important, they are given special names e and l respectively. These are given by:

Let us also look at some more important properties of this Schwarzschild geometry:

  1. Mass – The mass of the spherical star is the source of curvature of spacetime. The curvature is characterized by only the mass of the spherical star, and not by the distribution of the mass in the star.
  2. Schwarzschild radius – this is a quantity characterized by r = 2GM/c2. This radius is always less than the actual radius of the star and is very important to black hole physics as we will discuss later on in the paper.

To simplify the metric of Schwarzschild geometry we will redefine many units in the form of geometrized units where all quantities are measures in terms of length. Appendix A of [3] has more information and rules on these units.

In these units, the line element is given as:

The Schwarzschild radius in these units comes out to be: r = 2M.

Next, for another example, let’s look at the apparent change in frequency of emitted light, as observed by a moving observer.

Gravitational Redshift:

Gravitational redshift is the apparent change in frequency of light emitted from a star when observed from an observer far away.

Let the original frequency of the omitted light be w* and the frequency heard by an observer at infinity be w∞.

The energy of a photon is given by: E = hw.

To calculate the apparent frequency change we will use the concept of killing vectors to find the conserved quantity.

The conserved quantity is ξ.p where p is the four-momentum.

We also know that the energy is given by E = -p.uobs.

Therefore hw = -p.uobs

Also, uobs.uobs = -1 (normalization)

Since the spatial components of u is 0 for a stationary observer, the time component of the velocity is given by uobs  = (1-M/R)-1/2 using the metric given above.

ξ = (1,0,0,0)

Hence, uobs =  (1-M/R)-1/2. ξ

The Schwarzschild radius in these units comes out to be: r = 2M.

Next, for another example, let’s look at the apparent change in frequency of emitted light, as observed by a moving observer.

Gravitational Redshift:

Gravitational redshift is the apparent change in frequency of light emitted from a star when observed from an observer far away.

Let the original frequency of the omitted light be w* and the frequency heard by an observer at infinity be w∞.

The energy of a photon is given by: E = hw.

To calculate the apparent frequency change we will use the concept of killing vectors to find the conserved quantity.

The conserved quantity is ξ.p where p is the four-momentum.

We also know that the energy is given by E = -p.uobs.

Therefore hw = -p.uobs

Also, uobs.uobs = -1 (normalization)

Since the spatial components of u is 0 for a stationary observer, the time component of the velocity is given by uobs  = (1-M/R)-1/2 using the metric given above.

ξ = (1,0,0,0)

Hence, uobs =  (1-M/R)-1/2. ξ

Finally we plug in this value into the energy equivalence equation and obtain that:

Finally we plug in this value into the energy equivalence equation and obtain that:

This equation shows the apparent decrease in frequency seen by a stationary observer far away that tilts the color of the star towards the red end of the spectrum.

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