Relativitas Umum Matematis

Azas Ekuivalensi dalam Bentuk Matematis

Dalam relativitas umum, gravitasi tidak lagi dipikirkan hanya sebagai gaya. Gravitasi telah dipikirkan sebagai sebuah kelengkungan ruang-waktu. Relativitas umum menghasilkan relativitas khusus sebagai sebuah pendekatan yang konsisten dengan azas ekuivalensi. Jika kita fokuskan perhatian kita pada wilayah cukup kecil ruang-waktu, maka wilayah ruang-waktu tersebut dapat dipandang tidak mempunyai kelengkungan dan akibatnya tidak ada gravitasi. Meskipun kita tidak dapat menghilangkan medan gravitasi melalui suatu transformasi secara global, kita dapat semakin dekat ke kerangka acuan inersial ideal jika kita buat laboratoriumnya semakin kecil dalam volume ruang-waktu. Dalam laboratorium yang tengah jatuh bebas (tak-berotasi) yang menempati wilayah kecil ruang-waktu, hukum-hukum fisika adalah hukum relativitas khusus. Oleh karena itu dapat diharapkan semua persamaan relativitas khusus bekerja dengan baik di segmen cukup kecil ruang-waktu.

Dalam relativitas khusus ungkapan invarian yang mendefinisikan waktu proper, , dalam suku invarian panjang  4-D  sadalah

(1.1)

di mana

(1.2)

Di sini kita menggunakan perjanjian jumlahan Einstein dengan dan bergerak dari  0 ke 3 mengkaitkan  ke koordinat ruang-waktu umum . Koordinat 4-D  terkait dengan koordinat standard ruang dan waktu cT, X, Y, Z dengan cara berikut ini:

(1.3)

Penjelasan bagi ungkapan dalam persamaan  (1.1) adalah sebagai berikut. Panjang  4-D antara dua peristiwa yang terpisahkan oleh jarak infinitisimal diberikan oleh.

(1.4)

Pola ini mendefinisikan syarat yang harus dipenuhi oleh transformasi kerangka acuan relativistik agar dapat disebagai sebagai sebua transformasi yang absah. Sebarang transformasi yang tetap menjaga bentuk ini (invarian) disebut sebagai Tranformasi Lorentz. Let us assume that there is a frame where the two events occur at the same space location but at different times. This frame is said to be the proper frame for these events. The time interval between the two events in this frame is said to be the proper time . In this proper frame the expression (1.4) must take the form

(1.5)

Hence, expression (1.1) is just a statement that the 4-D length in the proper frame and the nonproper frame (having coordinates cT, X, Y, Z) must be equal. The matrix (1.2) has been used in (1.1) to make the expression more succinct and general. This general matrix form will prove to be very useful later in the context of general relativity.

Now we analyze what occurs in more general coordinate systems. Say we change the coordinates to

(1.6)

.

We allow this function of the coordinates to be any arbitrary function. Since it's arbitrary we no longer will be in an inertial frame of reference necessarily.

The invariant 4-D length equation now takes the new form:

(1.7)

where  is some connecting array. Therefore, from 4-D length invariance

(1.8)

Using the derivative relation

(1.9)

we get an expression for the quantity :

(1.10)

In the freely falling frame we must have the motion equation:

(1.11)

if we consider the motion of a free particle. Just as long as the free-float frame in a limited region of space lasts this equation must be in effect. In terms of the arbitrary coordinates  that we introduced above we can use the chain rule to obtain a new motion equation :

(1.12)

where we have defined the  quantity as

(1.13)

Note that equation (1.12) can be rewritten in a form that corresponds with the Newtonian 2^nd law of motion when an external force is present:

(1.14)

.

We have introduced the mass of the moving object here as . The messy part that came in when we switched to arbitrary coordinates can be viewed as a force if we insist on thinking of ourselves as being in a simple inertial frame where (1.11) is in effect in a force free situation. Such coordinate-dependent forces are called fictitious forces.

(1.15)

Note that the fictitious force is proportional to the mass of the object of attention. Einstein noticed that this was very much like the way that the mass enters into standard Newtonian gravitational theory.

(1.16)

As far as we know, no other physical forces are proportional to mass like the gravity force or the fictitious forces are. Hence, Einstein thought deeply about what was similar and what was different between a fictitious force and the gravitational force. This thinking led to the more general theory of relativity that he had set out to find after his success with the theory of special relativity.

Einstein's proposals for general relativity were as follows. Proper time in general relativity (defined in terms of any coordinates  ) is given by

(1.17)

and the equation of motion in general relativity is given by

(1.18)

.

In contrast, however, to special relativity,  no preferred coordinates in general relativity that will reduce these two equations to the form , and , globally. The array  is taken to represent the gravitational field. Although locally you can always manage to make this quantity look like the Minkowski metric (1.2), globally it can never be equivalent to the Minkowski metric since the Minkowski metric implies a spacetime that has no curvature, and therefore has no gravity.

Tensors

A Manifold is a set of points that locally looks like a bit of Euclidean n-dimensional space. N real coordinates may label the points in this manifold. The manifold can support a differentiable structure. This means that the functions involved in changes of coordinates are differentiable. A manifold differs from a surface in that it can stand alone as a mathematical structure. A surface, on the other hand, is going to be embedded in a higher dimensional Euclidean space.

Example: Compare a 2-sphere  (surface of a 3-D sphere) manifold with the fundamental Euclidean plane  manifold.

                             compared to

Even though the 2-sphere is a compact manifold, locally it has Euclidean character. Hence, locally it's no different from the  manifold, which goes off to infinity. The topology of the structures is not relevant since a topological statement is a global statement.

Although parts of manifolds can be covered by coordinate systems it is not the case that you can get a suitable coordinate system to cover the entire manifold. What we want is a coordinate system to cover the manifold uniquely. If it doesn't do so then the coordinate system is said to be a degenerate coordinate system. An example of a degenerate coordinate system is the 2-D polar coordinates . When , the variable  is indeterminate. Despite this shortcoming, polar coordinates are used often since they are very useful in certain situations.

To get around problems with coordinate systems at various points of a manifold we adopt a strategy of using more than one set of coordinates. We use the minimum number of distinct set of coordinates that will successfully cover all of the points of the manifold.

The manifold is said to be 'patched' by the different coordinate sets. The set of all necessary coordinate patches is called an atlas. Since manifolds are differentiable structures it is possible to go from one coordinate system patch to another by using overlapping regions of the patches. 

In general relativity we want to make global statements about physics independent of the local coordinate patches that are necessary to cover the 4-D spacetime manifold. In order to do this Einstein used tensors to represent physical quantities. Equations written in terms of tensors automatically maintain the same pattern amongst the tensor quantities no matter what coordinate system is being referred to.

We will now introduce the mathematical technology necessary for writing equations in terms of tensors. Consider the following change of coordinates in a n-dimensional manifold.

(2.1)

This change of coordinates can be written more succinctly as

(2.2)

where  denote the n functions , and  represents the old coordinates . To get information about how one coordinate system changes with the other we can construct the Jacobian transformation matrix, which gives the following.

(2.3)

The Jacobian, formed from this matrix, is defined as the determinant of the matrix in (2.3) :

(2.4)

We assume that for the range of coordinates that we are considering, J is well defined. In that case the inverse transformation can be solved for. It follows from the product rule for determinants that, if we define the Jacobian of the inverse transformation by

(2.5)

then .

Contravariant tensors

Consider two neighbouring points in the manifold: P and Q . Let these points have coordinates  and , respectively.  The two points define an infinitesimal displacement or infinitesimal vector .  The vector is not to be regarded as free, but as being attached to the point P . 

                           

The components of this vector in the  -coordinate system, are  which are connected to the  by

(2.6)

                                                   

A contravariant tensor of rank 1 is a set of quantities, written  in the  -coordinate system, associated with a point P, which transforms under a change of coordinates according to

(2.7)

where the transformation matrix is evaluated at P.  The infinitesimal vector  is a special case of (2.7) where the components  are infinitesimal. An example of a vector with finite components is provided by the tangent vector  to the curve .  It is important to distinguish between the actual geometric object like the tangent vector in the following diagram (depicted by an arrow)

                                           

and its representation in a particular coordinate system, like the n numbers  in the  -coordinate system and the different numbers  in the  -coordinate system.

We now generalize the definition (2.7) to obtain contravariant tensors of higher rank.  A contravariant tensor of rank 2 is a set of n² quantities associated with a point P, denoted by  in the  -coordinate system, which transform according to

(2.8)

The quantities  are the components in the  -coordinate system, the transformation matrices are evaluated at P, and the law involves two dummy indices  and .  An example of such a quantity is provided by the product , say, of two contravariant vectors  and .  The definition of third and higher-order contravariant tensors proceeds in an analogous manner.  An important case is a tensor of rank 0, called a scalar or scalar invariant , which, at P, transforms according to

(2.9)

.

Covariant and mixed tensors

As in the last section we begin by considering the transformation of a prototype quantity.  Let

(2.10)

be a real-valued function on the manifold, i.e. At every point P in the manifold  produces a real number.  We also assume that  is continuous and differentiable, so that we can obtain the differential coefficients .

Now, using the inverse of the coordinate transformation, the equation (2.10) can be written equivalently as

(2.11)

Differentiating this with respect to , using the function-of-a-function rule, we obtain

(2.12)

Then changing the order of the terms, the dummy index, and the free index (from  to  ) gives

(2.13)

This is the prototype equation for covariant tensors. Notice that it involves the inverse transformation matrix .  Thus, a covariant tensor of rank 1 is a set of quantities, written  in the  -coordinate system, associated with a point P, which transforms according to

(2.14)

Again, the transformation matrix occurring is assumed to be evaluated at P.  Similarily, we define a covariant tensor of rank 2 by the transformation law

(2.15)

and so on for higher-rank tensors.  Note the conversion that contravariant tensors have raised indices whereas covariant tensors have lowered indices.  (The way to remember this is that co goes below.)  The fact that the differentials  transform as a contravariant vector explains the convention that the coordinates themselves are written as  rather than , although note that it is only the differentials and not the coordinates which have tensorial character.

We can go on to define mixed tensors in the obvious way.  For example, a mixed tensor of rank 3- one contravariant rank and two covariant rank-- satisfies

(2.16)

If a mixed tensor has contravariant rank p and covariant rank q, then it is said to have type (p , q).

We now come to the reason why tensors are important in mathematical physics.  Let us illustrate the reason by way of an example.  Suppose we find in one coordinate system that two tensors, X_ab and Y_ab say, are equal, i.e.

(2.17)

Let us multiply both sides by the matrices  and  and take the implied summations to get

(2.18)

Since  and  are both covariant tensors of rank 2 it follows that .  In other words, the equation (2.17) holds in any other coordinate system.  In short, a tensor equation that holds in one coordinate system necessarily holds in all coordinate systems.  Thus, although we need to introduce coordinate systems for convenience in tackling particular problems, we wish to work with tensorial equations that are coordinate-independent so that our physics relationships remain unchanged

Using Tensors

Elementary Operations with Tensors

Tensor operations are operations on tensors that result in quantities that are still tensors.  A simple way of establishing whether or not a quantity is a tensor, is to see how it transforms under a coordinate transformation.  For example, we can deduce directly from the transformation law that two tensors of the same type can be added together to give a tensor of the same type, e.g.

(2.1.1)

                                      

The same holds true for subtraction and scalar multiplication.

A covariant tensor of rank 2 is said to be symmetric if

(2.1.2)

                                                 ,

in which case it has only  independently components (check this by establishing how many independent components there are of a symmetric matrix of order n).  A similar definition holds for a contravariant tensor .  The tensor  is said to be anti-symmetric or skew symmetric if

(2.1.3)

                                              ,

which has only  independently components; this is again a tensorial property.  A notation frequently used to denote the symmetric part of a tensor is

(2.1.4)

                     

and the anti-symmetric part is

(2.1.5)

                    

In general the symmetrization of a tensor relative to its covariant indices can be written:

(2.1.6)

In general the antisymmetrization of a tensor relative to its covariant indices can be written:(2.1.7)

For example, consider the covariant rank 3 antisymmetric tensor

(2.1.8) .

(A way to remember the above expression is to note that the positive terms are obtained by cycling the indices to the right and the corresponding negative terms by flipping the last two indices).  A totally symmetric tensor is defined to be one equal to its symmetric part, and a totally anti-symmetric tensor is one equal to its anti-symmetric part.

We can multiply two tensors of type  and  together and obtain a tensor of type , e.g.

(2.1.9)                                

In particular, a tensor of type  when multiplied by a scalar field  is again a tensor of type .  Given a tensor of mixed type , we can form a tensor of type  by the process of contraction, which simply involves setting a raised and lowered index equal.  For example,

(2.1.10)

i.e. A tensor of type  has become a tensor of type .  Notice that we can contract a tensor by multiplying by the Kronecker-Delta tensor , e.g.

(2.1.11)                           

In effect, multiplying by  turns the index  into  (or equivalently the index  into  ).

Using Tensors

Elementary Operations with Tensors

Tensor operations are operations on tensors that result in quantities that are still tensors.  A simple way of establishing whether or not a quantity is a tensor, is to see how it transforms under a coordinate transformation.  For example, we can deduce directly from the transformation law that two tensors of the same type can be added together to give a tensor of the same type, e.g.

(2.1.1)

                                      

The same holds true for subtraction and scalar multiplication.

A covariant tensor of rank 2 is said to be symmetric if

(2.1.2)

                                                 ,

in which case it has only  independently components (check this by establishing how many independent components there are of a symmetric matrix of order n).  A similar definition holds for a contravariant tensor .  The tensor  is said to be anti-symmetric or skew symmetric if

(2.1.3)

                                              ,

which has only  independently components; this is again a tensorial property.  A notation frequently used to denote the symmetric part of a tensor is

(2.1.4)

                     

and the anti-symmetric part is

(2.1.5)

                    

In general the symmetrization of a tensor relative to its covariant indices can be written:

(2.1.6)

In general the antisymmetrization of a tensor relative to its covariant indices can be written:(2.1.7

For example, consider the covariant rank 3 antisymmetric tensor

(2.1.8) .

(A way to remember the above expression is to note that the positive terms are obtained by cycling the indices to the right and the corresponding negative terms by flipping the last two indices).  A totally symmetric tensor is defined to be one equal to its symmetric part, and a totally anti-symmetric tensor is one equal to its anti-symmetric part.

We can multiply two tensors of type  and  together and obtain a tensor of type , e.g.

(2.1.9)                                

In particular, a tensor of type  when multiplied by a scalar field  is again a tensor of type .  Given a tensor of mixed type , we can form a tensor of type  by the process of contraction, which simply involves setting a raised and lowered index equal.  For example,

(2.1.10)

i.e. A tensor of type  has become a tensor of type .  Notice that we can contract a tensor by multiplying by the Kronecker-Delta tensor , e.g.

(2.1.11)                           

In effect, multiplying by  turns the index  into  (or equivalently the index  into  ).

Tensor Calculus

Partial Derivative of a Tensor

Partial differentiation of a tensor is in general not a tensor.  Depending on the circumstance, we will represent the partial derivative of a tensor in the following way

(3.1)

where we have taken the special case of a contravariant vector . 

We now show explicitly that the partial derivative of a contravariant vector cannot be a tensor. Consider the transformation relation for such a tensor.

(3.2)

Differentiating with respect to coordinate , we find

(3.3)

Using the chain rule this becomes:

(3.4)

Expanding this out we get:

(3.5)

           

If only the first term on the right-hand side were present, then this would be the usual tensor transformation law for a tensor of type (1,1).  However, the presence of the second term prevents  from behaving like a tensor.

This problem arises because of the very definition of the derivative. Differentiation involves comparing a quantity evaluated at two neighbouring points, P and Q say, dividing by some parameter representing the separation of P and Q, and then taking the limit as this parameter goes to zero.  In the case of a contravariant vector field , this would involve computing

(3.6)

for some appropriate parameter .  However, from the transformation law

in the form ,

(3.7)

we see that

(3.8)

and

(3.9)

This involves the transformation matrix evaluated at different points! Thus it is clear that  is not a tensor.  Similar remarks hold for general rank tensor differentiation.

To define a tensor derivative we shall introduce a quantity called an affine connection and use it to define covariant differentiation.  We will then introduce a tensor called a metric and from it build a special affine connection, called the metric connection, and again we will define covariant differentiation but relative to this specific connection.

The Affine Connection and Covariant Differentiation

Consider a contravariant vector field  evaluated at a point Q, with coordinates , near to a point P, with coordinates .  Then, by Taylor's theorem,

(3.10)

to first order.  If we denote the second term by , i.e.

(3.11)

then  is not tensorial since it involves subtracting tensors evaluated at two different points.  We are going to define a tensorial derivative by introducing a vector at Q that in some general sense is 'parallel' to  at P.  Since  is close to , we can assume that the parallel vector only differs from  by a small amount, which we denote .                                               

                                               

By the same argument as in previous discussion of the partial derivative,  is not tensorial, but we shall construct it in such a way as to make the difference vector

(3.12)

tensorial.  It is natural to require that  should vanish whenever  or  does.  Then the simplest definition is to assume that  is linear in both  or , which means that there exist multiplicative factors  where

(3.13)

and the minus sign is introduced to agree with convention.

We have therefore introduced a set of  functions  on the manifold, whose transformation properties have yet to be determined.  This we do by defining the covariant derivative of , (usually written in one of the following notations  ) by the limiting process

(3.14)

In other words, it is the difference between the vector  and the vector at Q that is still parallel to , divided by the coordinate differences, in the limit as these differences tend to zero.  Using (3.10) and (3.13), we find

(3.15)                                   

or in terms of the semi-colon notation

(3.16)                                   

Note that in the formula the differentiation index  comes second in the downstairs indices of .  If we now demand that  is a tensor of type (1,1), then a straightforward calculation (exercise) reveals that  must transform according to

(3.17)                                                           

or equivalently (exercise)

(3.18)

.

If the second term on the right-hand side were absent, then this would be the usual transformation law for a tensor of type (1,2).  However, the presence of the second term reveals that the transformation law is linear inhomogeneous. (3.17) or (3.18) is called an affine connection [or sometimes simply a connection or affinity].  A manifold with a continuous connection prescribed on it is called an affine manifold.  From another point of view, the existence of the inhomogeneous term in the transformation law is not surprising if we are to define a tensorial derivative, since its role is to compensate for the second term that occurs in (3.5).

We next define the covariant derivative of a scalar field to be the same as its partial derivative, i.e.

(3.19)

If we now demand that covariant differentiation satisfies the usual product rule of calculus, then we find

(3.20)

Notice again that the differentiation index comes last in the  -term and that this term enters with a minus sign.  The name covariant derivative stems from the fact that the derivative of a tensor of type (p, q) is of type (p, q+1), i.e. it has one extra covariant rank.  The expression in the case of a general tensor is:

(3.21)

It follows directly from the transformation laws that the sum of two connections is not a connection or a tensor.  However, the difference of two connections is a tensor of type (1,2), because the inhomogeneous term cancels out in the transformation.  For the same reason, the anti-symmetric part of a , namely,

(3.22)

is a tensor (called the torsion tensor).  If the torsion tensor vanishes, then the connection is symmetric, i.e.

(3.23)

Affine Geodesics

If  is any tensor, then we introduce the notation

(3.24)

that is,  of a tensor is its covariant derivative contracted with  .  A contravariant vector field  determines a local congruence of curves,

whenever the tangent vector field to the congruence is

We next define the absolute derivative of a tensor  along a curve C of this congruence,

written ,

by the following relation

(3.25)

The tensor  is said to be parallely propagated, or parallel transported, along the curve C if

(3.26)

This is a first-order ordinary differential equation for , and so given an initial value for , say , equation (3.26) determines a tensor along C which is everywhere 'parallel' to .

Using this notation, an affine geodesic is defined as a privileged curve along which the tangent vector is propagated parallel to itself.  In other words, the parallely propagated vector at any point of the curve is parallel, that is, proportional  to the tangent vector at that point:

(3.27)

Using (3.25), the equation for an affine geodesic can be written in the form

(3.28)

or equivalently (exercise)

(3.29)

The last result is very important and so we shall establish it afresh from first principles using the notation of the last section.  Let the neighbouring points P and Q on C be given by  and

(3.30)

to first order in , respectively.  This is essentially a Taylor expansion. We define

(3.31)

.

The vector  at P is now the tangent vector .  The vector at Q parallel to  is, by (3.13) and(3.31),

(3.32)

The vector already at Q is

(3.33)

to first order in .  These last two vectors must be parallel, so we require

(3.34)

where we have written the proportionality factor as  without loss of generality, since the equation must hold in the limit .  Subtracting produces the equation we obtained before:

(3.35)

  

Note that  appears in the equation multiplied by the symmetric quantity , and so even if we had not assumed that  was symmetric the equation picks out its symmetric part only.

If the curve is parameterized in such a way that  vanishes (that is, by the above, so that the tangent vector is transported into itself), then the parameter is a privileged parameter called an affine parameter, often conventionally denoted by s, and the affine geodesic equation reduces to

(3.36)

or equivalently

(3.37)

An affine parameter s is only defined up to an affine transformation (exercise)

(3.38)

where  and  are constants.  We can use the affine parameter s to define the affine length of the geodesic between two points  and  by , and so we can compare lengths on the same geodesic.  However, we cannot compare lengths on different geodesics (without a metric) because of the arbitrariness in the parameter s.  From the existence and uniqueness theorem for ordinary differential equations, it follows that corresponding to every direction at a point there is a unique geodesic passing through the point as shown below. 

                                                                   

Similarly, as long as the points are sufficiently 'close', any point can be joined to any other point by a unique geodesic.  However, in the large, geodesics may focus, that is, meet again as shown in the following diagram.

 

The Riemann Curvature Tensor and Geodesic Coordinates

Riemann Tensor

Covariant differentiation, unlike partial differentiation, is not in general commutative.  For any tensor , we define its commutator to be

(5.1)

Let us work out the commutator in the case of a vector .  Using the definition for covariant differentiation of a contravariant rank one tensor we see that

(5.2)

This is a tensor of mixed tensor of type (1,1). Taking the covariant derivative once again we get

(5.3)                                                          

with a similar expression for , namely,

(5.4)

      Subtracting these last two equations and assuming that

(5.5)          

we obtain the result

(5.6) where  is defined by

(5.7)          

Moreover, since we are only interested in torsion-free connections, the last term in (5.6) vanishes. Using the notation for antisymmetric tensors we get can rewrite (5.6) as follows:

(5.8)

Since the left-hand side of (5.8) is a tensor, it follows that  is a tensor of type (1,3).  It is called the Riemann tensor.  It can be shown that, for a symmetric connection, the commutator of any tensor can be expressed in terms of the tensor itself and the Riemann tensor.  Thus, the vanishing of the Riemann tensor is a necessary and sufficient condition for the vanishing of the commutator of any tensor. 

Geodesic Coordinates

We now prove a very useful result.  At any point P in a manifold, we can introduce a special coordinate system, called a geodesic coordinate system, in which

(5.9)

Here we are using a particular coordinate system so we use the notation where equal signs have an asterisk * above them to indicate that the result is not general but is wholly reliant upon the characteristics of the coordinate system we evaluate with respect to.

We can, without loss of generality, choose P to be at the origin of coordinates  and consider a transformation to a new coordinate system

(5.10)

           

where  are constants to be determined.  Differentiating (5.10), we get

(5.11)

(5.12)

      Then, since  vanishes at P, we have

(5.13)

from which it follows immediately that the inverse matrix

(5.14)

      We can now use the above results in the affine connection transformation law (3.17)

(5.15)       ,

We find the following relation between the affine connection in the two coordinate systems:

(5.16)

      Since the connection is symmetric, we can choose the constants so that

(5.17)

and hence we obtain the promised result

(5.18)

Many tensorial equations can be established most easily in geodesic coordinates.  Note that, although the connection vanishes at P, the derivative of the affine connection may not.

(5.19)

It can be shown that the result can be extended to obtain a coordinate system in which the connection vanishes along a curve, but not in general over the whole manifold.  If, however, there exists a special coordinate system in which the connection vanishes everywhere, then the manifold is called affine flat or simply flat.  This is intimately connected with the vanishing of the Riemann tensor. The following theorem holds in this respect for Riemann tensors.

Theorem: A necessary and sufficient condition for a manifold to be affine flat is that the Riemann tensor vanishes.

[For a proof of this theorem see section 6.7 of the book Introducing Einstein's Relativity by Ray d'Inverno.]

06/10/2004 12:17 PM

  

 

Introduction to the Metric

Metric Fundamentals

Any symmetric covariant tensor field of rank 2, say , defines a metric.  A manifold endowed with a metric is called a Riemannian manifold.  A metric can be used to define distances and lengths of vectors.  The infinitesimal distance (or interval in relativity), which we call , between two neighbouring points  and  is defined by

(6.1)          

Note that this gives the square of the infinitesimal distance, . This is conventionally written as .  The equation (6.1) is also known as the line element.  The square of the length of a contravariant vector  is defined by

(6.2)

The metric is said to be positive definite or negative definite if, for all vectors ,  or , respectively.  Otherwise, the metric is called indefinite.  The angle between two vectors  and  with  and  is given by

     

(6.3)                                                                                

      In particular, the vectors  and  are said to be orthogonal if

(6.4)

If the metric is indefinite (as in relativity theory), then there exist vectors that are orthogonal to themselves called null vectors, i.e.

(6.5)

      The determinant of the metric is denoted by

(6.6)

     

The metric is non-singular if , in which case the inverse of , , is given by the relation

(6.7)

.

It follows from this definition that gab is a contravariant tensor or rank 2 and it is called the contravariant metric.  We may now use  and  to lower and raise tensorial indices by defining

(6.8)

and

(6.9)

where we use the same kernel letter for the tensor.  Since from now on we shall be working with a manifold endowed with a metric, we shall regard such associated contravariant and covariant tensors as representations of the same geometric object.  Thus, in particular, , , and  may all be thought of as different representations of the same geometric object, the metric g.  Since we can raise and lower indices freely with the metric, we must be careful about the order in which we write contravariant and covariant indices.  For example,  could possibly be different from .

Metric Geodesics

Consider the time like curve C with parametric equation .  Dividing equation (6.1), which we write as

(6.10)

,

by the square of  we get

(6.11)

.

Then the interval s between two points P₁ and P₂ on C is given by

(6.12)

We define a timelike metric geodesic between any two points P₁ and P₂ as the privileged curve joining them whose interval is stationary under small variations that vanish at the end points.  Hence, the interval may be a maximum, a minimum, or a saddle point.  Deriving the geodesic equations involves the calculus of variations and the use of the Euler-Lagrange equations. The Euler-Lagrange equations result in the second-order differential equations

(6.13)

where the quantities in curly brackets are called the Christoffel symbols of the first kind and are defined in terms of derivatives of the metric by

(6.14)

            Multiplying through by  and using  we get the equations

(6.15)

where  are Christoffel symbols of the second kind defined by

(6.16)

In addition, the norm of the tangent vector  is given by (6.11).  If, in particular, we choose a parameter u which is linearly related to the interval s, that is,

(6.17)                

where  and  are constants, then the right-hand side of (6.15) vanishes.  In the special case when , the equation for a metric geodesic becomes

(6.18)

and

(6.19)

where we assume .

Apart from trivial sign changes, similar results apply for spacelike geodesics, except that we replace s by , say, where

(6.20)

We would have in this case

(6.21)

and

(6.22)

where we assume .

However, in the case of an indefinite metric, there exist geodesics, called null geodesics, for which the distance between any two points is zero.  It can also be shown that these curves can be parameterized by a special parameter u, called an affine parameter, such that their equation does not possess a right-hand side, that is,

(6.23)                                                           

where

(6.24)

The last equation follows since the distance between any two points is zero, or equivalently the tangent vector is null.  Again, any other affine parameter is related to u by the transformation

(6.25)

where  and  are constants.

The Metric Connection

            In general, if we have a manifold endowed with both an affine connection and metric, then it possesses two classes of curves, affine geodesics and metric geodesics, which will be different as shown in the diagram below where affine geodesics are in the up/down direction and the metric geodesics are in the right/left direction. 

However, comparing the two curve equations (3.37) and (6.17):

we see that the two classes of curves will coincide if we take

(6.26)

or, using (6.16) and (6.14), if

(6.27)

It follows from the last equation that this special connection based on the metric is necessarily symmetric, i.e.

(6.28)

If one checks the transformation properties of  from first principles, it does indeed transform like a connection (exercise).  This special connection built out of the metric and the metric's derivatives is called the metric connection.  From now on, we shall always work with the metric connection and we shall denote it by  rather than , where  is defined by (6.27).  This definition leads immediately to the identity (exercise)

(6.29)

.

Conversely, if we require that (6.29) holds for an arbitrary symmetric connection, then it can be deduced (exercise) that the connection is necessarily the metric connection.  Thus, we have the following important result.

Theorem:  If  denotes the covariant derivative defined in terms of the affine connection , then the necessary and sufficient condition for the covariant derivative of the metric to vanish is that the connection is the metric connection.

            In addition, we can show that

(6.30)

and

(6.31)

.

Metric Induced Curvature

The Riemann curvature tensor (or Riemann-Christoffel tensor) is defined in terms of the connection by the relation,

(7.1)    

where  is the metric connection, given as

(7.2)   

Thus,  depends on the metric and its first and second derivatives.

At any point P of a manifold  is a symmetric matrix of real numbers.  Therefore, by standard matrix theory, there exists a transformation which reduces the matrix to diagonal form with every diagonal term either +1 or -1. 

The excess of plus signs over minus signs in this form is called the signature of the metric.  Assuming that the manifold is continuous and non-singular, the signature is an invariant.  In general, it will not be possible to find a coordinate system in which the metric reduces to this diagonal form everywhere.  If, however, there does exist a coordinate system in which the metric reduces to diagonal form with  1 diagonal element everywhere, then the metric is called flat.

How does metric flatness relate to affine flatness in the case we are interested in, that is, when the connection is the metric connection?  The answer is contained in the following result.

Theorem:  A necessary and sufficient condition for a metric to be flat is that its Riemann tensor vanishes.

Necessary Condition Discussion:

            Necessity follows from the fact that there exists a coordinate system in which the metric is diagonal with  1 diagonal element.  Since the metric is constant everywhere, its partial derivatives vanish and therefore the metric connection  vanishes as a consequence of the definition (7.2).  Since  vanishes everywhere, then so must its derivatives.  The Riemann tensor therefore vanishes by the definition (7.1).

Sufficient Condition Discussion:

            Since we are using the metric connection, we know that

(7.3) 

This can be expanded to give the relation

(7.4)  

from which we get

(7.5)

If the Riemann tensor vanishes, then by the Riemann curvature theorem concerning affine connections that was discussed in section 5, we know that there exists a special coordinate system in which the  connection vanishes everywhere.  From equation (7.5) it follows that

(7.6)

.

This means that the metric must be constant everywhere. Hence, it can be transformed into diagonal form with diagonal elements  1.  Note that the result (7.5) expresses the ordinary derivatives of the metric in terms of the connection.  This equation will prove useful.

            Combining this metric-induced Riemann curvature theorem with the Riemann curvature theorem concerning affine connections, we see that if we use the metric connection then metric flatness coincides with affine flatness.

It follows immediately from the definition of the Riemann tensor

(7.7)

that it is anti-symmetric on its last pair of indices :

(7.8)

The fact that the connection is symmetric leads to the identity

(7.9)

Lowering the first index with the metric, then it is easy to establish, for example by using geodesic coordinates, that the lowered tensor is symmetric under interchange of the first and last pair of indices, that is,

(7.10)

            Combining this with equation (7.8), we see that the lowered tensor is anti-symmetric on its first pair of indices as well:

(7.11)        

            Collecting these symmetries together, we see that the lowered curvature tensor satisfies

(7.12)

(7.13)

These symmetries considerably reduce the number of independent components; in fact, in n-dimensions, the number is reduced as follows:

(7.14)

.

In addition to the algebraic identities, it can be shown, again most easily by using geodesic coordinates, that the curvature tensor satisfies a set of differential identities called the Bianchi identities:

(7.15)

Ricci Tensor

            We can use the curvature tensor to define several other important tensors.  The Ricci tensor is defined by the contraction

(7.16)

Since

(7.17)

then we see that the Ricci tensor is symmetric.

(7.18)

which by (7.16) is symmetric. 

Ricci Scalar

A final contraction defines the Ricci scalar R by

(7.19)

Einstein Tensor

These two tensors can be used to define the Einstein tensor

(7.20)

which is also symmetric.

Contracted Bianchi Identities

            By (7.15), the Einstein tensor can be shown to satisfy the contracted Bianchi identities

(7.21)

Note that in different General Relativity textbooks authors will adopt different sign conventions for how the curvature tensor, and its associated contracted forms depend on the affine connections. In such books the Riemann tensor or the Ricci tensor can have the opposite signs to the definitions given above.

Notation: The book Schaum's Outline - Tensor Calculus by David Kay uses an unusual definition for the partial derivative of the metric. Kay uses the following definition.

and then also uses the uncommon definition

.

10/10/2002 12:20 PM

 

 Spacetime Dimensions and the Weyl Tensor

            The algebraic identities

(8.1) 

and

(8.2)    

lead to the following special cases for the curvature tensor:

We get this from the symmetry relation in the form

or in the following alternate form straight from the definition

(8.3)

So in 2-D the Riemann tensor is proportional to the Ricci scalar.

(8.4)

Note that a 3-D space where  necessarily makes the Riemann tensor zero in 3-D. As we will see later a zero Ricci tensor in 4-D general relativity does not imply  and this in turn implies the existence of a nonzero gravitational field. Hence, from the above relation we have obtained the result that in 3-D, a zero Ricci tensor condition does imply that  and that therefore the 2-D gravitational field must be zero.

(8.5)

This last equation can be generalized to n-dimensions when . This generalization gives the following result

(8.6)

The Weyl tensor (or conformal tensor) is defined to be the tensor . In n-dimensions, with , the Weyl tensor can be written as follows.

(8.7)

            In four dimensions, we have

(8.8)

            It is straightforward to show that the Weyl tensor possesses the same symmetries as the Riemann tensor, namely,

(8.9) 

and

(8.10) 

However, it possesses an important extra symmetry

(8.11)

Combining this result with the previous symmetries, it then follows that the Weyl tensor is trace-free, in other words, it vanishes for any pair of contracted indices.  One can think of the Weyl tensor as that part of the curvature tensor for which all contradictions vanish.

            Two metrics  and  said to be conformal to each other or (conformally related ) if

(8.12)

where  is a non-zero differentiable function.  Given a manifold with two metrics defined on it, which are conformal, then it is straightforward from (8.12) to show that angles between vectors are the same for each metric.  This is shown as follows. Let  and  be two rank-one tensors. The angle between the vectors is defined through the relation

(8.13)

Using (8.12) gives the following

(8.14)

Ratios of magnitudes of vectors also remain invariant under conformal transformations. Moreover, the null geodesics of one metric coincide with the null geodesics of the other (exercise).  The metrics also possess the same Weyl tensor, i.e.

(8.15)

Any quantity that satisfies a relationship like (8.15) is called conformally invariant (gab, , and  are examples of quantities which are not conformally invariant).  A metric is said to be conformally flat if it can be reduced to the form

(8.16)

where  is a flat metric (the special relativity 'Minkowski' metric). 

We end this section by quoting two results concerning conformally flat metrics.

Theorem:  A necessary and sufficient condition for a metric to be conformally flat is that its Weyl tensor vanishes everywhere.

Theorem:  Any two-dimensional Riemannian manifold is conformally flat.

Notes on the Weyl Tensor

The Weyl tensor in General Relativity provides curvature to the spacetime when the Ricci tensor is zero. In General Relativity the source of the Ricci tensor is the energy-momentum of the local matter distribution. If the matter distribution is zero then the Ricci tensor will be zero. However the spacetime is not necessarily flat in this case since the Weyl tensor contributes curvature to the Riemann curvature tensor and so the gravitational field is not zero in spacetime void situations. This term allows gravity to propagate in regions where there is no matter/energy source.

07/02/2005 4:54 PM

 

Invariant Integrals and Tensor Densities

We would like to be able to integrate a quantity over a particular range of coordinate values in such a way that the integrand gives the same value in any other generalized coordinate system. If the integrand is a pure scalar quantity, then this is easily achieved because of the way that scalar quantities transform. Let  be such a pure scalar quantity. The transformation law that it obeys when a new primed coordinate system is introduced is as follows:

(9.1)

Say that we are considering the scalar quantity  at two distinct points . We can sum the scalar function evaluated at the two distinct points in a new primed coordinate system such that the following relation holds:

(9.2)

Here  are the same points in the new coordinate system. Unlike tensors of higher rank, a scalar field can be evaluated at two different points and still be a scalar field.  There should be no problem making this behavior hold if we go to infinitesimal sums.

(9.3)

When we are integrating over coordinate ranges we want the following integral invariance to hold.

(9.4)

The word 'Quantity' is meant to represent a tensor of some general type. However, with integrands of this pattern we run into a problem in that what we are integrating is not necessarily a pure tensor quantity. We know that the 4-D volume element  transforms according to the following Jacobian relation

(9.5)

.

This relation implies that the differential element  transforms in a funny way. It's not transforming like a scalar quantity and it's not transforming like a vector quantity. This differential element transforms according to a rule that's similar to tensors but is different in that a power of the transformation Jacobian comes into the transformation. A new set of geometric quantities called Tensor Densities can be defined in an analogous manner to tensors but the transformations involve powers of the transformation Jacobian. A general definition of the tensor density can be written in the following way.

Tensor density: A tensor density,  , of weight  transforms like a tensor except that the W^th power of the Jacobian appears as a factor with the pattern shown below.

(9.6)

Since the differential element  transforms according to equation (9.5) with the pattern

(9.7)

then  must be a scalar density of weight . The integrand in (9.4) will be a scalar only if the factor labelled 'Quantity' is a tensor density of weight . To make integrals be independent of the coordinates, the integrand is multiplied by the square root of the metric determinant as shown in the following expression.

                                                                       

(9.8)

This works since the metric transforms according to

(9.9)

Since the right hand side of this equation is essentially the product of three matrices multiplied together, we can use the rule for the product of matrix determinants to give

(9.10)

The value g is negative for an indefinite metric so when we take the square root of this relation we insert a minus sign and the result is

(9.11)

Thus  is a scalar density of weight . We then see that the integral invariance given by (9.8) works since we have made

(9.12)

and therefore (9.3) must be realized.

The covariant derivative of a tensor density has the following pattern

(9.13)

For example, the covariant derivative of a vector density  has the form

(9.14)

For the special case when  this leads to the important divergence equation

(9.15)

In terms of a tensor density formed from multiplying a tensor  by , this divergence expression becomes

(9.16)

It can be shown that the metric determinant, which acts as a scalar density of weight 2, satisfies the following relations.

(9.17)

and

(9.18)

Relations (9.16), (9.17), and (9.18) turn out to be of great use in the Lagrangian formulation of general relativity.

15/10/2002 3:07 PM

 

How spacetime curvature and coordinate transformations affect the metric connection  and the Riemann tensor .