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We extend standard linear perturbations of a Schwarzschild black hole by Chandrasekhar to Bondi frame by transforming both even and odd parity perturbations when angular momentum l = 2.

In studying linear perturbations of a Schwarzschild black hole we are able to study its static space-time properties and the emission of gravitation radiation. The gravitational radiation emitted by a Schwarzschild black hole carries information about its mass (as well as spin and charge for rotating and/or charged black holes). Also by studying the perturbations of a Schwarzschild black hole it is possible to make conclusions about the stability of the Einstein equations [

In this paper we transform well-known linear perturbations of a Schwarzschild black hole to Bondi-Sachs form. The outline of this paper is as follows: in Section 2 we discuss the Bondi-Sachs formalism as background material. In Section 3, we discuss linearized Bondi-Sachs metric. In Section 4, we discuss the complex notation to be used. In Section 5, we transform the linear perturbations of a Schwarzschild black hole to Bondi-Sachs frame. Section 6 is a discussion. The paper ends with the conclusion in Section 7.

We use coordinates based upon a family of outgoing null hypersurfaces

where

We work in spherical polar coordinates

We now introduce the complex dyad

We also introduce the complex quantities U, J defined by

and

For spherically symmetric case (Schwarzschild space-time), we take J = 0 and U = 0. J and U are interlinked, and they contain all the dynamic content of the gravitational filed in the linearized regime [

We define the operator

which has the property of raising the spin-weight by 1, and similarly we define

which has the property of lowering the spin-weight by 1.

For a Schwarzschild space-time, we have

We linearize Bondi-Sachs metric in order to find J, U,

where a and b are functions of r and

From Equation (3) we have

From Equation (4) we have

Lastly

The spherical harmonics

At this stage we must deal with a notational issue concerning the use of complex numbers to represent physical quantities. J and U are complex and are used as a convenient representation of metric quantities with two real components. However, it is also common practice to represent oscillations in time as

with

Not only is the above a more compact notation, but also it is much easier to manipulate

The difficulty is that the complex nature of J and U on the one hand, and of

The above construction was not made in [

The general metric for time-dependent axisymmetric systems in general coordinates

where

where

When the Schwarzschild metric is perturbed we have_{2}, and q_{3} are taken as quantities of the first order of smallness (for odd-parity perturbations) as it is the case with

We start by transforming t to u by performing the following transformation

where

Then we substitute Equation (26) into the perturbed metric and we chose a function ^{2} zero to the zeroth order in^{2} is zero to 1st order in

and

After the above transformation, we note that

where

We substitute Equation (30) into the transformed metric and apply the condition that the coefficient of

We found

where

Finally, we transform r to a new r' by performing the following transformation

were

We use Equation (35) to find

After the above transformation, we found the transformed metric to be given by

which simplifies to

By comparing the transformed even-parity metric perturbations with the linearized Bondi-Sachs metric (see Section 2) and noticing that

or

By substituting functions (27), (28), (32), and (36) into Equations (47), (48), (49), and (50),

where

where

where

where

We have used the trigonometric identities:

to simplify

The expressions for

From Equation (23) we have

since

We start by transforming t to u by the following transformation

where

By substituting Equation (66) into Equation (61) and choosing the function ^{2} zero to the zeroth order in

where a function

We then transform

where

Then by substituting Equation (70) into Equation (67) and choosing

where a function

where

After the above transformation procedure, we found the transformed metric to be

Substituting Equation (72) in the above metric components, they simplify to

By comparing the transformed odd-parity metric perturbations with the linearized Bondi-Sachs metric (see Section 2) we found that

From Equation (17) with

From Equation (16) with

The expressions for J and U obtained above involve both complex quantities i and j. Taking the real part with respect to i leads to

and

Thus, both U and J are pure imaginary quantities.

The transformation of linear perturbations of a Schwarzschild black hole to Bondi-Sachs is complete. The transformation of even-parity perturbations was much more involved than that of odd-parity perturbations. The end results of the transformation processes for both even and odd-parity perturbations were very different, for example, in the case of odd-parity perturbations, w and

All unknown functions;

It appears that the transformation of second order perturbations of a Schwarzschild black hole to Bondi-Sachs form will be extremely difficult to do. In the future, the extension of the work of this paper to a stationary charged (Reissener-Nordström) black hole will be very exciting and hopefully attainable. Similarly, the transformation of linear perturbations(gravitational) of a Kerr black hole will be very exciting to do, but the transformation of its standard metric to Bondi-Sachs form has been obtained only very recently [

ASK and NTB would like to thank the University of South African, Rhodes University, and National Research Foundation of South Africa for the financial support.