Generating Solutions to Einstein’s Equations by Conformal Transformationsklotza/KlotzUndergradThesis.pdf · · 2011-08-30Generating Solutions to Einstein’s Equations by Conformal

Generating Solutions to Einstein’s Equations by Conformal

Transformations

Alex Klotz

Department of Physics, Queen’s University

Honours Thesis

Supervisor: Dr. Kayll Lake

March 26, 2008

Abstract

New pefect fluid solutions to Einstein’s equations were generated using a method known as

conformal transformations. This method has been used previously to show relationships between

metrics, but here it is used to develop new and unique solutions. The metric 1

1− 2m(r)r

dr2 + r2dΩ2 −

e2Φ(r)dt2 was multiplied by a transformation function f(r) and perfect fluid conditions were imposed

to create a differential equation for the mass distribution m(r). Two transformation functions that

lead to a physically realistic mass distribution were found: f(r) = (1 + r2)−1/3 and f(r) = (1 +

r2)−1+1/√

2. The latter can be generalized to a family of transformation functions: f(r) = (r2 +

α)−1+1/2√−4 N+2 N2+4.

Dear reader: I wrote this short thesis as an undergraduate. Although it appears on Google Scholar, it has not been peer reviewed and is not guaranteed to meet any standards of scientific rigour. I have left it here for your interest; see Lake, K., Physical Review D v 77 i 12, 2008 for a more complete paper.

1 Introduction

General relativity is a modern theory of gravity that describes how massive bodies warp

space and time around them. More formally, the distribution of geodesics, the paths

of least action, through spacetime is related to the distribution of energy. The mathe-

matical relationship that quantifies general relativity are the Einstein equations. These

state that the Einstein tensor G, which dictates the gravitational attraction and contains

information about geodesic paths, is directly proportional to the stress-energy tensor T,

which contains information about the distribution of energy and momentum through

spacetime.

Gba = 8πT ba (1)

Both these tensors are 4x4 matrices with rows and columns corresponding in spherical co-

ordinates, from left to right and top to bottom, to time, azimuthal angle, polar angle, and

radius. The significance of this equation is that all the information about the structure of

spacetime is given essentially by the distribution of mass and energy. Exact solutions to

the Einstein equations are necessary to model astrophysical phenomena ranging from the

interior of neutron stars to the formation of galactic haloes. This theory is not restricted

to the realm of mathematics; the Einstein equations have practical use. For example, the

GPS system initially failed because the software did not take into account the effects of

general relativity.

A perfect fluid is a region in which the stress energy tensor is diagonal; the time

component represents the energy density, and the three spacial components are identical

and represent the isotropic pressure. An example of a perfect fluid solution to Einstein’s

equation is the Schwarzschild Interior solution, which describes the spacetime of the

interior of a spherical star of constant density. This solution is, however, physically

unrealistic because of its infinite speed of sound propagation.

The Einstein Tensor or Stress Energy Tensor of a perfect fluid looks as follows:

1


ρ 0 0 0

0 p 0 0

0 0 p 0

0 0 0 p

(2)

Each spacetime can be defined by a metric, a 4x4 matrix that contains elements used

to construct a line element. A metric can be transformed into a new one by multiplying

it by a function f(r). This is known as a conformal transformation. The Einstein tensor

based on this metric can also be a perfect fluid if the right conditions are satisfied. By

multiplying a familiar metric by a transformation function, new perfect solutions can be

generated. There are currently a limited number of known perfect fluid solutions, and

the purpose of this thesis is to detail a method that can greatly increase that number.

There are several simplifactions made in this paper. So called “natural” units are used

to simplify expressions, meaning that quantities such as c, the speed of light, and G, the

gravitational constant, are set equal to unity. A restriction placed on metrics is that

they are static, meaning independent of time (as opposed to, for instance, the metric of

a rotating body), and spherically symmetric, meaning there is no angular dependence.

General relativity is known for having very complicated mathematical expressions

that are difficult for a person to analyze. Fortunately, a software package known as

GRTensor, which organizes and performs operations on the various tensors involved in

general relativity, was developed by Peter Musgrave, Denis Pollney and Kayll Lake.

GRTensor is used in this paper for this purpose, specifically as an extension of Maple, a

computer algebra software.

2 Literature Review

This topic dates back to 1970, when H.A. Buchdall proved the conformal flatness of the

Schwarzschild interior solution in a paper of the same name [1]. The process is described

whereby Minkowski space can be transformed into the Schwarzschild interior solution.

2


This was one of the first papers that demonstrated that it was possible to conformally

transform from one metric to another and produce a physically interesting result.

A particularly relevant piece of literature relating to this topic is one of similar name

and focus, “Use of Conformal Mapping to Construct New Solutions to the Einstein Equa-

tions” by A.V. Nosovets, published in the Soviet Union in the 1970s and not cited until

now [6]. The method used in this thesis is discussed and a rough derivation is given.

Several examples are provided in which various time-dependent functions are given to

yield commonly known perfect fluid solutions. Also discussed is the form of the Ein-

stein tensor under conformal transformation. The method of conformal transformations

is generally known amongst general relativity scholars and discussed in such books as

Exact Solutions of Einstein’s Equations by Stephani et al.[7]. It again covers the basic

mathematics but states that no new solutions have been found with this method. The

same feature is common among most literature on this topic; the method is discussed,

but no examples leading to new solutions are given.

It was proven by Lake and Delgaty [3] that, of over 100 established percect fluid

metrics, all but 16 proved to be physically unrealistic. This issue was resolved in a highly

relevant paper, “All static spherically symmetric perfect fluid solutions of Einstein’s

equations”, by Kayll Lake [4]. This paper proves that an infinite multitude of perfect

fluid solutions to Einstein’s equations exists, which was unknown prior to its publication.

The fact that an infinite number of solutions exists means that it is definitely possible to

discover new ones. It also provides the metric that is the basis of this research.

3 Theoretical Basis

The description of conformal transformations is a simple one. Consider a static spheri-

cally symmetric metric in spacetime:

3


g =

A(r) 0 0 0

0 r2 0 0

0 0 r2sin2θ 0

0 0 0 B(r)

(3)

(3)

The associated line element can be written as:

ds2 = A(r)dr2 + r2dΩ2 −B(r)dt2 (4)

where dΩ is the square angle area element dΩ2 = dθ2 + sin2θdφ2 A metric g is said to be

a conformal transformation of g if

g = f(r)g (5)

and

˜ds2 = f(r)ds2 = f(r)A(r)dr2 + f(r)r2 sin2 θdΩ2 − f(r)B(r)dt2 (6)

where f(r) is known as the transformation function. It is known that a simple perfect

fluid solution, the Schwarzschild interior solution, is conformally flat [1]. This means

that the Schwarzschild interior solution can be reached by a conformal transformation

of Minkowski space. However, it is more recently known that conformal flatness is not

necessary to generate a perfect fluid. For instance, if the metric ds2 = dr2 + dω2 −

(1 + r2/s2)2dt2 is transformed by the function 1/(1 + r2/R2)2, the resulting metric is the

Schwarzschild interior solution. The initial metric is not a perfect fluid, indicating that

a perfect fluid can be generated from a metric that is not.

Based on work by Nosovets[6], Geroch [2], and Professor Lake, it has been derived

that the Einstein tensor under conformal transformation takes the form

4


Gab = Gab −∇a∇b ln f(r) + 12dda

ln f(r) ddb

ln f(r)

+gab(gab∇ab ln f(r) + 1/4gab 1

4dda

ln f(r) ddb

ln f(r)(7)

At this point it becomes simpler to use GRTensor to evaluate these quantities. For

instance, the transformed radial component of the Einstein tensor can be expressed as:1

f (r)Grr +

1

4 (f (r))3A (r)B (r) r2×

3(ddrf (r)

)2B (r) r2 + 2 f (r) r2

(ddrB (r)

)ddrf (r) + 8 f (r)B (r) r d

drf (r)

+4 (f (r))2B (r)− 4 (f (r))2A (r)B (r) + 4 r (f (r))2 ddrB (r)− 4 (f (r))4 r d

drB (r)

+4 (f (r))4A (r)B (r)− 4 (f (r))4B (r)

(8)It

is of note that the transformed tensor still contains elements from the metric (as opposed

to just the Einstein tensor); thus, there is no expression that can directly describe a

stress-energy tensor under transformation.

4 Uniqueness of New Metrics

When a new metric is generated by conformal transformation the concern arises as to

whether it is different from the original, or merely a coordinate transformation. To test

this, it is advantageous to examine the quantities in general relativity that are invariant

under transformation. One of the simplest quantities to examine is the Ricci scalar.

The transformed Ricci scalar has factors of both f(r) and f’(r), a change that cannot be

explained by mere coordinate transformation.

As an example, the Ricci scalar of Minkowski space is zero, but when transformed by

f(r)=r, the Ricci scalar is −9/(2r3).

Another quantity that is invariant under transformation is the ratio of the Ricci scalar

to the square of the Weyl tensor at the point where the pressure vanishes (the boundary

of the star). In the specific case of the metrics and functions tested in this project,

1This is a single equation, not a matrix or vector. It is read top-left to bottom-right. The same applies to other largeequations in this paper.

5


this quantity is indeed different between pre- and post-transformed metrics, for example,

going from 4.2 to 12.1 after a conformal transformation.

From these tests it appears that the new metrics are indeed new, and not merely

coordinate transformations of the original.

5 The Lake Metric

The metric used primarily in this paper comes from a 2003 paper by my supervisor Dr.

Lake [4]. The metric in curvature coordinates is:

ds2 =1

1− 2m(r)r

dr2 + r2dΩ2 + e2Φ(r)dt2 (9)

The term m(r) is the radial mass distribution of the fluid. The constraint that is placed

on m(r) is:

m (r) =

e

∫ 3−2 ( ddr

Φ(r))2r2+3 ( d

drΦ(r))r−2 r2 d2

dr2 Φ(r)

r(( ddr

Φ(r))r+1)dr

×

∫re

∫ 2 r2 d2

dr2 Φ(r)+2 ( ddr

Φ(r))2r2−3 ( d

drΦ(r))r−3

r(( ddr

Φ(r))r+1)dr (

r d2

dr2 Φ (r) +(

ddr

Φ (r))2r − d

drΦ (r)

)((ddr

Φ (r))r + 1

)−1dr + C

(10)

Φ is chosen, based on various conditions and constraints, to be

Φ(r) =N

2ln(1 +

r2

α) (11)

N is an integer greater than zero, and alpha is any positive constant. This is the definition

of Φ used in this paper, but other definitions can be used. The metric is static and

spherically symmetric, and different perfect fluids can be described based on the choice

of f(r) and the parameters for Φ. The mass distribution m(r) is obtained based on the

choice of Φ. The majority of currently known perfect fluid solutions can be expressed

in the form of this metric with values of N from 1 to 5. It has already been used by

Lattimer and Prakash [5] with N=2 to describe the interior of a neutron star.

The reason that this metric is used for this paper is that when it is conformally

6


transformed, the components of its Einstein tensor contain terms of both m(r) and f(r),

and the two can be related by a differential equation. Solving that differential equation

allows the mass distribution of the fluid to be expressed given the transformation function.

6 A Word on Mathematics

Maple was used to perform mathematical functions such as integration, differential equa-

tion solving, and simplification. The GRTensor extension was also used. It was often

the case that a mass distribution would contain an integral that Maple would leave in

unevaluated form. When this happened, the expression could be copied and converted

into Mathematica notation, and pasted into the Wolfram Online Integrator2. This web-

site can perform integrals that Maple cannot, and was indispensable in this project. The

solved integrals can then be converted back into Maple notation and pasted into the

expressions. I differentiated some of the more complicated integrals that Wolfram gave

me, and I got the same expression back.

Certain functions yielded integrals that could not be solved in Maple or Mathematica.

A common feature among these is an integral of an exponential of an inverse hyperbolic

tangent of a polynomial. Other functions gave results that were expressed as a function

of the sum of roots of a polynomial. I could not evaluate functions in this form. These

expressions are not necessarily physically meaningless, but they cannot be explored in

this project.

It is necessary to note that mathematically complicated or unlikely functions do not

necessarily discount physical reality. For instance, if the transformation function (r2 +

1)−3/2 is used, the resulting mass distribution contains factors of i, and the constant of

integration necessary for positive pressure is complex. The values of all the quantities,

however, are real.

2http://integrals.wolfram.com/index.jsp

7


7 Role of N and alpha

The time component of the metric is given as e2Φ(r), where N2ln(1 + r2

α). This evaluates

to −(1 + r2/α)N . For the untransformed case, given a nonzero constant of integration,

an increasing N decreases the rate at which the mass function diverges. The role of α

is more difficult to observe graphically, but it also acts as a scaling factor. The effect of

both can be seen in figures 1 and 2.

Figure 1: Mass distributions with varying values of N

Figure 2: Mass distributions with varying values of alpha

If a function with a given N and alpha yields a desired result, then changing N or

8


alpha should yield a similar result. If the function yields a mass distribution that can

be easily evaluated (i.e. contains no integrals that Maple cannot perform), then success

in this matter is defined if changing N and alpha will still be easily evaluated by Maple.

This is not always the case.

8 Generation of Mass Distributions

A perfect fluid has a stress-energy tensor, and, equivalently, an Einstein tensor, with only

diagonal components. The time component of the tensor represents the energy density

of the fluid, while the other diagonal components are all equal to the isotropic pressure.

The Einstein tensor of the Lake metric under conformal transformation takes the form:

ρ 0 0 0

0 pΩ 0 0

0 0 pΩ 0

0 0 0 pr

(12)

Thus, in order to have a perfect fluid, pΩ = pr3. Or, more formally:

Grr − GΩ

Ω = 0 (13)

Using GRTensor, the Lake metric was imported with a given transformation function.

The Einstein tensor was generated and an expression was defined as the difference be-

tween the radial and angular components, resulting in a differential equation of m and r,

as mentioned earlier. The spacetime is a perfect fluid when these components are equal,

so the differential equation was solved for the difference equal to zero, resulting in a mass

distribution as a function of radius. The algorithm can be seen in Appendix B.

Transformation functions were chosen to be only functions of radius. This implies

that the spherical symmetry of the metric is conserved, and that only static metrics are

dealt with. Choosing transformation functions and observing the mass distribution was

3Ω can signify either θ or φ

9


the primary method of investigation.

9 Chain Rule for the Mass Distribution Algorithm

Ideally, if the resulting mass distribution for a given function f=f(r) is known, the mass

from a function G(f(r)) would be a minor variation of m. Unfortunately, this is not the

case. In general, the mass distributions are long and unwieldy, but this compounded

greatly when two functions are added to each other, or even if a constant is added to the

function.

It is difficult to predict what a mass distribution will look like based on the trans-

formation function. This is magnified by the fact that seemingly small changes in f(r)

can have drastic changes in m. For instance, the function f(r) = r2 + 1 yields a mass

distribution

m(r) = −r3 − 1/2 r5 +r3 C1 (1 + r2)

7/3

(4 r2 + 1)4/3(14)

However, a slight modification of the transformation function: f = r2 + 2 yields a mass

distribution

∫(7 r4 + 10 + 16 r2) r

√4 r4 + 7 r2 + 2e

317

√17arctanh(1/17 (8 r2+7)

√17) (1 + r2)

−2(2 + r2)

−4dr

(1 + r2) (2 + r2)3r3e−

317

√17arctanh(1/17 (8 r2+7)

√17) (4 r4 + 7 r2 + 2)

−3/2

(15)

which at present cannot be integrated. This is not to say that if the function could be

integrated it would necessarily be different from the first example, but the output of the

algorithm is different and much more complicated. This is an example of how minor

changes in the transformation function can cause huge changes in the mass distribution.

If a function f(r) leads to a certain mass distribution m(r), then a scalar multiple,

f(r) = kf(r) will yield the exact same mass distribution: it is invariant under scalar

multiplication.

10


10 Mass Distributions for Various Functions

Different transformation functions f(r) were inputted into the aforementioned algorithm

to generate m. I will summarize some of the results. For simplicity’s sake these cases

were evaluated with N and alpha both equal to 1.

The simplest transformation is f = 1, the untransformed case. This results in

m(r) =(1 + r2) r3 C1

2 r2 + 1+ 1/2

r3

2 r2 + 1, (16)

which increases with r. There is one constant of integration in a given mass distribution.

This constant is crucial in allowing new solutions to be physically realistic.

Power functions, f(r) = rn lead to a mass distribution that has different forms for

different n.

For positive even values, r takes the form as a polynomial with odd-power terms from

r to rn+3. The coefficient of the linear term approaches 12

for increasing n. The higher

order terms get increasingly small. At high even n, the mass function is nearly linear for

r < 1, but rapidly diverges to +infinity if n/2 is even and -infinity if n/2 is odd.

Figure 3: Mass distribution for f(r) equal to an even power of r. It is linear for x < 1 then diverges.

11


For positive odd exponents, the result is a rational expression multiplied by an integral:

∫ (r2n

[(n+ 2)r2 + n+ 1

]n+2)− 1

n+2

dr (17)

that could not be evaluated in Maple but results in the hypergeometric function. Negative

exponents were similar to their odd counterparts, as were rational exponents such as

square roots.

Some transformation functions yielded mass distributions that could not be calculated,

including exponentials, gaussians, and trigonometric functions.

Mass distributions often contain the hypergeometric function, an integral related to

the gamma function and used in topology, which is not unknown in general relativity.

Some other special functions that have manifested themselves in mass distributions are

the imaginary error function and the general beta function. Mass distributions containing

the hypergeometric function are often physically unrealistic.

These, of course, are just examples of the infinitude of functions that can be used.

Each one of these mass distributions represents a perfect fluid solution to Einstein’s

equations, although most will be physically unrealistic.

11 Physicality of Solutions

The conditions for a physically realistic solution are noted by Lake, 2003 as follows:

“(i)isotropy of the pressure, (ii)regularity at the origin, (iii) positivity of the pressure

and energy density at the origin, (iv)vanishing of the pressure at a finite boundary,

(v)monotone decrease of the energy density to the boundary, and (vi) subluminal adia-

batic sound speed.”[4]

The first condition, isotropy of pressure, requires that all the non-time diagonal com-

ponents of the Einstein tensor (equivalently, the pressure components of the stress-energy

tensor) be equal. This is biconditional with the fact that the new spacetime is a perfect

fluid.

12


The second condition stipulates that there is not a singularity at the origin. Regu-

larity is guaranteed by having a positive f(r) and a zero valued f’(r) at the origin. This

requirement is elaborated upon in Appendix A. If the transformation function does not

satisfy the aforementioned criteria, the density will be infinite at the origin, indicating a

singularity.

There are several criteria that a mass distribution and its associated pressure and

density must meet. Mass must obviously be zero at the origin and positive at all other

points. Furthermore, since the mass contained within a sphere cannot decrease as the

sphere increases in radius, its first derivative must also remain non-negative.

The density, defined as

ρ =1

4πr2

dm

dr(18)

must also remain positive. Variations or fluctuations of the density create an unstable

system, so it must have no turning points. It is also not physically stable to have density

increasing as r approaches infinity. These two restrictions are idealized by a function

that is finite at the origin and decreases, asymptotically approaching zero, described in

criteriae (iii) and (iv).

The pressure, defined as:

r(ddr

Φ (r))

(r − 2m (r))−m (r)

4π r3(19)

again must be positive. This definition is derived from the stress-energy tensor. It is

nonphysical to have pressure outwardly increasing, so it must also decrease from its

value at the origin. The pressure in this case can fall below zero, signaling the end of the

body of a star. The radius at which the pressure equals zero is the boundary of the star.

An example of pressure and density in a physically realistic perfect fluid can be seen

in figure 4. Examples of different in which ways the density can signify a physically

unrealistic case can be seen in figure 5.

13


Figure 4: Plot of density (left) and pressure (right) as a function of radius for a physically realisticperfect fluid. The density decreases asymptotically to zero and the pressure drops below zero at a finiteradius.

The square of the adiabatic speed of sound can be defined as

v2 = dp/dρ =dp/dr

dρ/dr(20)

Ideally this function would be less than one at all points. If this is not the case, it can

still be argued that sound speed is subluminal, although it would not act as an ideal gas.

12 The Constant of Integration

When a mass distribution is generated there is one constant of integration, given by

Maple as C1. This constant is crucial to create a physically realistic spacetime. For

example, in the simplest case of f(r)=1, the resulting mass is as mentioned in equation

16:

m(r) =(1 + r2) r3 C1

2 r2 + 1+ 1/2

r3

2 r2 + 1, (21)

If the constant of integration is set, for instance, to one, then the resulting pressure will

be negative. For a given distribution, either m, ρ, or p can be negative depending on the

constant. To determine what the constant should be to have positive values, it is useful

to find the value of a given quantity on the origin, which is expressed as a linear function

of the constant, and solving for the constant. Since there are multiple quantities involved

that may not have the same requirements, an often practical method is to solve for the

r=0 value of pressure and density to be equal, which usually ensures that all quantities

14


Figure 5: Examples of physically unrealistic solutions include singularities (infinite density at the origin),instabilities (fluctuating density) and negative density

are well behaved. Figure 6 gives an example of how to find an appropriate constant.

Since the mass is (hopefully) zero at the origin and the density (having the same sign

as the mass’ derivative) is positive, the mass will increase as expected. This method

works for most cases, although there are exceptions.

13 Determining f(r) for a perfect fluid

In a more general case, f can be a function of time and radius. If this type of function

is used to transform the metric, the resulting Einstein tensor may have off-diagonal

components at Grt and Gt

r. Assuming f(r,t) is separable, generating this in GRTensor

allows one to create a partial differential equation to determine what f(r,t) should be in

order to have a perfect fluid (i.e. the off diagonal components go to zero and the spacial

diagonal components are equal). Running this algorithm yields a time function that is

constant, eliminating the issue for static cases, and the radial equation f(r), assuming the

aforementioned expression of Φ is used, is

15


Figure 6: Pressure and density at the origin for the f(r)=1 mass distribution as a function of theintegration constant. In this example the system is well behaved for the range -0.5 to 0.5, with theequality point at -0.25.

√√√√2 rA (α+ r2)2 N hypergeom

([1/2, 1− 2N ], [3/2],−r

2

α

)α−1

((1 +

r2

α

)2 N)−1

+B((α+ r2

)N)−1

(22)

In the specific case where N=1 and α = 1, this simplifies to

1/3

√18 rA+ 6 r3A+ 9B

1 + r2(23)

This gives a family of mass distribution functions based on the choice of constants A and

B. If A is set to zero (with any N or alpha) then f(r) is a constant times 1/(r2 +α). The

pressure and density of this distribution are both constant; the Einstein tensor is merely

16


made up of numbers, dependent on the constant of integration:

Gab =

3 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

(24)

In the previous example, C1 = 1. In the A=0 case, the speed of sound in the fluid

would be 0/0, which is indeterminate. If the constant of integration is set to −1/2

then all the components become zero, and it is a vacuum solution. Since its Riemann

tensor has zero-valued components, it becomes Minkowski space and shows that the

Lake metric is conformally flat. For other values of A there is a more complicated term

in the denominator, the mass distribution of which currently cannot be evaluated. The

algorithm used to generate this function can be seen in Appendix D. Although this

algorithm only ensures that there are no off-diagonal components, the conditions for

perfect fluidity are imposed when the mass is generated.

14 The r2 + 1 Family of Functions

The process of running the algorithm with various transformation functions and exam-

ining the resulting mass can be time consuming and pointless, especially given software

limitations. However, I was fortunate to arbitrarily choose a function f(r) = r2 + 1 that

yields a mass function that is simple (less than four terms, no complicated integrals)

and that produces positive mass, density, and pressure near the origin. The reason that

these functions work well with the algorithm is that the time component of the metric

also takes the form (1 + r2), so multiplying this by the same f(r) does not complicate the

mass function significantly. In addition to this function, if f(r) is a positive integer power

of this function, f = (r2 + 1)n, the resulting mass is similar to the n=1 case. The mass

increases from the origin and then plummets, and the density starts from a positive value

17


and plummets, while the pressure increases to infinity. Although this fluid is physically

unrealistic, choices of n besides positive integers yield other interesting cases.

The first physically realistic case that I discovered was n=-1/3. After solving the

necessary equations for the constant of integration (approximately 0.61 in this case), the

mass distribution, density, and pressure all satisfied the aforementioned conditions. The

range of values of the constant to produce positive pressure and density at the origin is

approximately is about 0.36 to 1.35. This will be discussed in greater detail in section

16.

Another interesting case is f(r) = 1/√r2 + 1. The mass and pressure are similar to

the aforementioned case, but the density increases from the origin, reaches a maximum,

and decreases, asymptotically approaching zero. Although this satisfies all the conditions

of physicality, the turning point in the energy density is unstable. Modifying the constant

so that the derivative of the density is zero at the origin (i.e. the turning point is at r=0)

unfortunately causes a negative mass distribution.

Ideally, changing N or alpha would have a similar effect as in simpler cases. Simply

changing N or alpha from 1 leads to a more complicated functions with special integrals,

often resulting in a hypergeometric function with complex values at r > 0. If the (r2 + 1)

expression is modified, however, to (r2/α+ 1)−N , then the functions are again generated

with the same efficiency for any values as for 1. Unfortunately they do not behave

as nicely when the entire expression is raised to an exponent, as they do with N=1.

With regards to the reciprocal cube root case, changing alpha still generates a physically

realistic solution, but modifying N leads to a complex hypergeometric function.

15 Inverse Process

Most of my research has involved choosing a function f(r) and using it to transform a

metric and generate a mass distribution. Alternatively, one can input the desired mass

distribution and run a similar version of the algorithm to generate the function required

18


to yield such a mass. This is of particular use to astrophysicists who already know, be

it from measurement or derivation, the interior distribution of a given star and wish to

find a perfect fluid solution to Einstein’s equations to represent it.

The algorithm is similar (Appendix C), but the resulting algebra tends to be more

complicated. As an example, if the mass distribution for f=1 is inputted into this algo-

rithm, and the result is integrated (it contains an elliptical function of the second kind),

the resulting function is dependent on three constants (including one from initial func-

tion). When one of these constants is set to zero, the generated transformation function

becomes constant. Although the expected result was a constant, the elliptical function

was given. It is often very difficult to get meaningful results from this method, especially

as mass distributions become more complicated.

This process will add two constants to the generated transformation function. If an

initial mass distribution that was generated from the forwards process is given, modifying

the constants will often yield the original function. This method can in fact be useful: if

a function is known to have an associated mass distribution, that mass distribution can

be used to generate a family of functions, one of which is the original. Other choices of

constants may yield similar results when the mass is generated, but it is possible to gain

new information. As an example, using the mass from f(r) = 1/(r2 + 1) and generating

the function, in addition to the original function, it can yield 1/(2r2− 1). Obviously, the

resulting mass will not be much different.

Serendipity was again a factor when using this technique. When generating the func-

tion that would yield a vacuum spacetime (m=0), plugging the resulting function back

into the original algorithm and choosing an appropriate constant (the first constant equal

to zero) yielded another physically realistic solution. Again the transformation function

was from the same r2 + 1 family, with the exponent as −1 + 1/√

2. This satisfies all the

conditions including perfect fluidity.

The family of transformation functions where m(r)=0 scales with N and alpha. Alpha

only changes values within the function, but N changes the form more significantly. In

19


general, the function resulting from this method with a higher N or alpha can take the

form:

f(r) =(α + r2

)−1+1/2√−4N+2N2+4

(25)

The mass distribution for this function is:

m (r) =(r2N + α + r2

√−4N + 2N2 + 4− r2

)−−2+N+√−4 N+2 N2+4

N+√−4 N+2 N2+4−1 C1 r3 (26)

With the right choice of integration constant, the mass distributions can be physically

realistic for any choice of N or alpha. This family of solutions is the most signicant result

of this paper. 4

16 Properties of Physically Realistic Cases

As mentioned, two physically realistic cases were discovered: one represents a single

solution, while the other represents an entire family of solutions. The transformation

functions that yielded these were both of the form f(r) = (r2 +α)n, with n = −1/3 (the

reciprocal cube root) and n = −1 + 1/2√−4N + 2N2 + 4.

In the n=-1/3 case, N and α were equal to 1. The n=-1/3 case had a mass distribution:

m (r) =3

3√

1 + r2

(−1/8

(2√

3 arctan(1/3

(1 + 2 22/3 3

√1 + r2

)√3)

−2 ln(−1 + 22/3 3

√1 + r2

)+ ln

(1 + 22/3 3

√1 + r2 + 2

3√

2 (1 + r2)2/3 3√

2 + 3 C1 r3

(27)

The values for C1 are between 0.35 and 1.36 Plots of mass, pressure, and density show

that all the relevant conditions are satisfied. One criterion that is difficult to assess is

isotropy of pressure, which is a requirement for perfect fluidity. The Einstein tensor is

too large to be displayed, but the size of the expression is different for the radial and

angular components. This may indicate that the distribution is anisotropic in pressure

and therefore not a perfect fluid. However, when the difference between the radial and

4There may be more interesting results if different constants are chosen when generating f(r), but I was unable toevaluate them.

20


angular components is calculated for any point, it is always zero, and when their ratio

is calculated, it is always unity. Thus, the two expressions are equivalent but expressed

differently, most likely due to a simplification error. The speed of sound for this metric

was subluminal. Unfortunately, I was unable to generalize this case to all N.

The other physically realistic solution found was for n = −1 + 1/2√−4N + 2N2 + 4.

The mass distribution in this case is, as mentioned in equation 26:

m (r) =(r2N + α + r2

√−4N + 2N2 + 4− r2

)−−2+N+√−4 N+2 N2+4

N+√−4 N+2 N2+4−1 C1 r3 (28)

The first solution in this family was for N=1 and alpha=1, and the mass distribution in

this case is:

m(r) = 2C1 r3

(2 r2 +

√2)1/2

√2(

2 r2 +√

2)√

2√

2(29)

The ideal constant of integration in this case is approximately 0.22. The metric files

based on both of the aforementioned mass distributions can be seen in Appendix F, and

can be copied into their own file for future research.

In the N=1 case, the square of the speed of sound is nonzero at the origin and increases

towards c as r increases (figure 7). Throughout space the speed of sound is subluminal,

although it does get very close to the speed of light in the low density region far from

the origin. This condition is satisfied for N=1. However, for N > 1, the speed of sound

increases from its starting point, then decreases, then ceases to exist: the derivative of

the pressure has become positive; thus the square of the speed of sound is negative and

the speed is complex. The point where the speed of sound becomes complex is beyond

the point where the pressure reaches zero, meaning it is outside the star and not an issue.

Although the speed of sound is subluminal for N=1, it still increases as r approaches

infinity. Lake and Delgaty [3] state that only nine of the 16 physically realistic perfect

fluids have a speed of sound decreasing towards infinity, so it would be interesting to find

a mass distribution with such properties. More research is required in this area.

21


Figure 7: Speed of sound per c as a function radius for the irrational exponent case. It approaches thespeed of light but does not reach it.

Plotting m(r) for the untransformed metric and the new metrics shows three functions

of similar shape but increasing at different rates. The relationship between the divergence

can be explained as follows: at low r, they are all independent, but at higher r, the ratio

of the logarithms of each mass functions approach a constant. For example, at high r, the

mass of the n = 1/sqrt2− 1 case can be described as the mass of the untransformed case

to the power of approximately 0.88. The reason this is observed is that as r increases the

density approaches zero (required in order to be physically realistic), which is proportional

to the derivative over r2. All masses at high r thus approximate r3−ε where ε is very small,

so the masses all approximate similar functions at high r.

17 Discussion

It is apparent that this method can be used to generate physically realistic perfect fluid

solutions to Einstein’s equations. The transformation functions most likely to yield these

fluids take the form (r2 +α)n. I have come across the r2 +α family of functions by three

different methods: arbitrarily choosing functions and noting those that work, choosing

a mass distribution and observing the function that leads to it, and solving a differen-

22


tial equation to determine an ideal function. Two physically realistic cases that I have

generated are for n=-1/3 and n=1/√

2 − 1 for N=1; thus n is not restricted to rational

numbers. I have not yet had success with transcendental numbers. I have generalized

the latter of those functions to all N, creating a family of solutions infinite in size.

The fact that I was able to find more than one new solution in limited time indicates

that there are literally an infinite number of values that n can take, and a single person

cannot examine them all. However, algorithms can be produced to evaluate an entire set

of functions at once. A Maple worksheet that will generate all the mass distributions for

rational number exponents within a range can be seen in Appendix E. Similar algorithms

can be written for different types of numbers. This should be run in Mathematica so

that the integrals may be done more readily. For example, the integral

∫ r 3√

1 + r2

3 + 7 r2 + 4 r4dr (30)

cannot be solved by Maple; however, its integrated form appears in the mass distribution

for the n = −1/3 solution in the equation 27.

The findings of this paper are particularly significant because, as of 1998, there were

only 16 known physically realistic perfect fluid solution [3]. Professor Lake showed in 2003

that there are infinite perfect fluid solutions, and through my research, I have shown that

there is another infinite family entirely.

In addition to using Mathematica to perform integrals more readily, I believe this

research can be greatly improved if certain expressions, notably those that are expressed

as the sum of roots of a polynomial, are able to be evaluated. This would greatly increase

the amount of transformation functions that can be experimented. Perhaps an extension

to Maple can be written that can handle these functions, or a mathematician can analyse

them and provide a means of simplification. Among the first new functions to be tested is

that discussed when solving for an ideal function, the square root of a cubic polynomial

over 1 + r2 as in equation 23, and generalized to equation 22. Another inolves the

family of transformation functions that result in setting m(r) equal to zero, with both

23


constants nonzero. The mass distributions that arise from these families of functions

may be physically realistic, or, at the very least, an interesting topic of discussion.

The n=-1/3 case actually represents a family of solutions; each can be modified by

the choice of the constant and by changing alpha in Φ. Further work is required to

create an expression that is physically realistic with any choice of N. Further research

that can be done is repeating the process with a different choice of Φ or with a different

starting metric. It has been shown that non-perfect fluid solutions can act as a base for

transformation, so there are a vast number of possibilities in this area.

I have tested too many functions with interesting results to describe in this paper.

The masses of the two realistic perfect fluid families that I have discovered, as well as the

original untransformed metric, are all described with different families of functions. The

untransformed case is merely a polynomial, while one of the new ones involves irrational

exponents, and the other is described with logarithms and inverse tangents. Just based

on these few cases, it can be seen that there is great diversity in the fluids that can

potentially be developed.

18 Conclusion

The method of conformal transformations was used to discover two new families of phys-

ically realistic perfect fluid solutions to the Einstein equations, a significant amount

compared to the sixteen known in 1998. There is a definite possibility to generate new

and relevant perfect fluid solutions with this method. Future work in this area includes

improving the efficiency of the mathematical software used, attempting to find more

physically realistic solutions (including those with a radially decreasing sound speed),

using different variables for transformation (such as time and angle), using other choices

of Φ, and using other metrics as a starting point of transformation.

24


References

[1] H. A. Buchdahl. Conformal flatness of the schwarzschild interior solution. American

Journal of Physics, 39(2):158–162, 1971.

[2] R. Geroch. Asymptotic structure of space-time. In F.P. Esposito and L. Witten,

editors, Asymptotic Structure of Space-Time, pages 1–105, New York, U.S.A., 1977.

Plenum Press.

[3] K. Lake and M. Delgaty. Physical acceptability of isolated, static, spherically symmet-

ric, perfect fluid solutions of einstein’s equations. Computer Physics Communications,

115(2), December 1998.

[4] Kayll Lake. All static spherically symmetric perfect-fluid solutions of einstein’s equa-

tions. Phys. Rev. D, 67(10):104015, May 2003.

[5] James M. Lattimer and Madappa Prakash. Ultimate energy density of observable

cold baryonic matter. Physical Review Letters, 94(11):111101, 2005.

[6] A. V. Nosovets. Use of conformal, mapping to construct new solutions of the Einstein

equations. Soviet Physics Journal, 16:1572–1576, November 1973.

[7] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt. Exact So-

lutions of Einstein’s Field Equation. Cambridge University Press, Cambridge, U.K.,

New York, U.S.A., 2nd edition, 2003.

Appendix

A Proof of Boundary Conditions on f(r)

For any general spherically symmetric metric the line element can be written as

ds2 = A(r)dr2 +B(r)dΩ2 − C(r)dt2 (31)

Conditions on these functions indicate that if any of them is zero at the origin, there

25


is a singularity at that point [3]. Furthermore, the derivatives must be zero on the origin.

When the metric is transformed, the functions become A = (fA). If f(r) is zero on the

origin, then A is zero at the origin. Taking the derivative,

A′ = A′f + f ′A (32)

At r = 0, f(r) is nonzero and A’ is zero, so the remaining term is f’A. A is nonzero, so for

A′ to be zero at the origin, f’ must be zero at the origin. The full proof is too involved

to be repeated here; a full version can be seen in Lake and Delgaty.

B Algorithm for generating and examining mass distributions

This algorithm can be used in Maple with GRTensor installed with a user’s choice of

N, alpha, and f(r). It will produce graphs of mass, pressure, and density. Different

conditions may be applied to find a suitable constant of integration. A caveat: Maple

will likely encounter an integral that it will be unable to solve.

restart;

grtw():

N:=1;

alpha:=1;

Phi(r):=N/2*ln(1+r^2/alpha);

f(r):=(r^2+1)^(-1/3);

qload(staticsolvedcon1);

grcalc(G(dn,up));

gralter(_,1,6,7);

factor(grcomponent(G(dn,up),[r,r])-grcomponent(G(dn,up),[theta,theta]));

dsolve(%=0,m(r));

26


m(r):=rhs(%);

rho:=diff(m(r),r)/(4*Pi*r^2):

p:=(r*diff(Phi,r)*(r-2*m(r))-m(r))/(4*Pi*r^3):

_C1:=solve(limit(rho,r=0)-limit(p,r=0)=0);

plot(m(r),r=0..1);

plot(p,r=0..1);

plot(rho,r=0..1);

C Algorithm for generating transformation functions

This algorithm can be used in Maple with GRTensor installed to generate a transforma-

tion function that will yield an m(r) chosen by the user. Note that many mass functions

will often yield an insolvable expression.

restart;

grtw():

N:=1;

alpha:=1;


m(r):=r^3;

qload(staticsolvedcon1);

grcalc(G(dn,up));

gralter(_,1,6,7);

factor(grcomponent(G(dn,up),[r,r])-grcomponent(G(dn,up),[theta,theta]));

dsolve(%=0,f(r));

27


f(r):=rhs(f(r):=subs(C1 = 1,C 2 = 1, f(r));

D Algorithm to find perfect fluid transformation function

This algorithm conformal transforms the Lake metric by a function of radius and time

and determines the function that allows the off-diagonal components to vanish.

restart;

grtw();

grOptionTermSize:=1;

gr:=f(r,t)/(1-2*m(r)/r);

gth:=f(r,t)*r^2;

gp:=f(r,t)*r^2*sin(theta)^2;

gt:=-f(r,t)*exp(2*Phi(r));


qload(thesis_general1);

grcalc(G(dn,up));

gralter(_,1,6,7);

grdisplay(_);

pdsolve(grcomponent(G(dn,up),[t,r])=0,f(r,t));

pdsolve(grcomponent(G(dn,up),[r,t])=0,f(r,t));

dsolve(diff(_F1(r), r) = _c[1]/(_F1(r)*(alpha+r^2))-2*N*_F1(r)*r/(alpha+r^2));

If this is run in Maple then the resulting function for f(r) should be 1/3√

18 rA+6 r3A+9B1+r2

while f(t) should be constant.

28


E Algorithm for Generating Families of Mass Distributions

This algorithm can be used to generate mass distributions with the family f(r) = (r2+1)n

for n equal to all rational numbers within a given range. It uses two nested for loops; the

index variable of each loop is used as the denomenator or numerator of the fraction in

the exponent. The large block of code is the general form of m(r) as a function of f(r).

printlevel:=2;

for i from 1 by 1 to 3 do

for j from 1 by 1 to 3 do

f(r):=(r^2+1)^(i/j);

m(r)[i/j]:= simplify(

(int(-1/2*r*exp(-int((3*(diff(f(r), r))^2*r^6+3*

f(r)^2-2*f(r)*r^6*(diff(f(r),r,r))+3* (diff(f(r),

r))*r^5*f(r)-2*f(r)*r^2*(diff(f(r),r,r))+6*(diff

(f(r), r))^2*r^4+7*r^2*f(r)^2+3*(diff(f(r),

r))^2*r^2-4*f(r)*r^4*(diff(f(r),r,r))+6*f(r)^2*r^4+3*(

diff(f(r), r))*r*f(r)+6*(diff(f(r),

r))*r^3*f(r))/(r*f(r)*(2*r^4*f(r)+3*f(r)*r^2+(diff(f(r), r))

*r^5+ (diff(f(r), r))*r+2*(diff(f(r), r))*r^3+f(r))),

r))*(-2*f(r) *r^5*(diff(f(r),r,r))+2*f(r)*(diff(f(r), r))

*r^4-4*f(r)*r^3*(diff(f(r),r,r))+3*(diff(f(r),

r))^2*r^5+3*(diff(f(r), r))^2*r+4*f(r)*(diff(f(r), r))

*r^2+2*f(r)*(diff(f(r), r))+6*(diff(f(r),

r))^2*r^3+2*r^3*f(r)^2-2*f(r)*r*(diff(f(r),r,r)))/(f(r)*

(1+r^2)*((diff(f(r), r))*r^3+2*f(r)*r^2+(diff(f(r),

r))*r+f(r))), r)+_C1)*exp(int((3*(diff(f(r), r))^2 *r^

6+3*f(r)^2-2*f(r)*r^6*(diff(f(r),r,r))+3*(diff(f(r),

r))*r^5*f(r)-2*f(r)*r^2*(diff(f(r),r,r))+6*(diff(f(r), r))

^2*r^4+7*r^2*f(r)^2+3*(diff(f(r),

r))^2*r^2-4*f(r)*r^4*(diff(f(r),r,r))+6*f(r)^2*r^4+3*(di

ff(f(r), r))*r*f(r)+6*(diff(f(r),

r))*r^3*f(r))/(r*f(r)*(2*r^4*f(r)+3*f(r)*r^2+(diff(f(r), r))

*r^5+(diff(f(r), r))*r+2*(diff(f(r), r))*r^3+f(r))), r))

> );

29


> end do;

>

> end do;

F GRTensor Metric Files

Here is the metric file for the Lake metric as well as the two metric files I have created

for the physically realistic fluids I created. In each case some parameters, such as C1

may need modification.

This is the original Lake metric.

Ndim_ := 4 :

x1_ := r :

x2_ := theta :

x3_ := phi :

x4_ := t :

g11_ := f(r)/(1-2*m(r)/r) :

g22_ := f(r)*r^2 :

g33_ := f(r)*r^2*sin(theta)^2 :

g44_ := -f(r)*exp(2*Phi(r)) :

This is the metric for the family of functions mentioned. Assign values to N and alpha within maple.

Ndim_ := 4 :

x1_ := r :

x2_ := theta :

x3_ := phi :

x4_ := t :

_C1 := 1:

m(r):=r^3*(1+r^2*N+r^2*(4-4*N+2*N^2)^(1/2)-r^2)^(-(N+(4-4*N+2*N^2)^(1/2)-2)/(N+(4-4*N+2*N^2)^(1/2)-1))*_C1:

f(r):=(1+r^2)^(-1+1/2*(4-4*N+2*N^2)^(1/2)):

Phi(r):=N/2*ln(alpha+r^2):

g11_ := f(r)/(1-2*m(r)/r) :

g22_ := f(r)*r^2 :

30


g33_ := f(r)*r^2*sin(theta)^2 :

g44_ := -f(r)*exp(2*Phi(r)) :

This is the metric for the n=-1/3 case.

Ndim_ := 4 :

x1_ := r :

x2_ := theta :

x3_ := phi :

x4_ := t :

_C1 :=0.61:

m(r):=1/3*(-(2*sqrt(3)*arctan( (1 + 2*2^(2/3)*(1 + r^2)^(1/3))/ sqrt(3)) - 2*ln(-1 + 2^(2/3)*(1 + r^2)^ (1/3)) +

ln(1 + 2^(2/3)*(1 + r^2)^(1/3) + 2*2^(1/3)*(1 + r^2)^(2/3)))/ (4*2^(2/3)) +3*_C1)*r^3/(1+r^2)^(1/3):

Phi(r):=1/2*ln(1+r^2):

g11_ := f(r)/(1-2*m(r)/r) :

g22_ := f(r)*r^2 :

g33_ := f(r)*r^2*sin(theta)^2 :

g44_ := -f(r)*exp(2*Phi(r)) :

31


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Generating Solutions to Einstein’s Equations by Conformal Transformationsklotza/KlotzUndergradThesis.pdf · · 2011-08-30Generating Solutions to Einstein’s Equations by Conformal