1
Freefall in curved space
Contents:
1.Curvilinear coordinate systems
1.1.Non orthogonal example
1.2.General curvilinear systems
2.Tensors
3.Christoffel symbols, 1st and 2nd kind
3.1.Christoffel symbols as Derivatives of the Metric Tensor
4.Covariant derivative
5.The geodesic equation
6.Appendix A, a wakeup problem
7.Appendix B, index notation
8.Appendix C, remarks of interest
9.Appendix D, T-shirt from CERN
10.References
Abstract
This article derives the equation for a geodesic in a curved space. It assumes that the reader has a working knowledge of partial derivatives, the chain rule and some basics about vector algebra/calculus and variational calculus. It tries to introduce the concepts and notations that are necessary to be able to read and understand the different terms of the equation. Here is the ‘layout’ of the article:
-short about curvilinear coordinate systems
-some facts about tensors and why tensors are of fundamental importance in physics
-Christoffel symbols
-covariant derivative of a covariant and a contravariant vector
-derivation of the equation for a geodesic in curved space
As an introduction the reader is invited to look at a simple example in Appendix A and if unfamiliar with index notation also read Appendix B. Appendix C and D contains miscellaneous information about mathematically related areas.
1.Curvilinear coordinate systems
A useful observation :
= is a scalar function in 3 dimensions. For each point in space it has a scalar value. Setting = const.= c1 will introduce a dependency between the variables and thereby define a surface in the 3d space. A point on the surface is represented by a vector. A small variation of under the constraint = c1 will give a new point on the surface represented by the vector . The vector will clearly “lie in surface”. The situation is shown in the figure below.
d(c1) = 0 = =>
the vector is to the surface = c1
Anexample concerning dependence:
In 3 dimensional space where the cross product of vectors is defined, two functions and are not independent.
There exists a function = const. = c1 that connects the two. A necessary and sufficient condition is that = 0.
0 = => and are parallel => = 0 . This last expression is the condition for the determinant of a Jacobian matrix to be zero. This interdependence condition is also used in the Legendre transform.
1.1.Non orthogonal example
We start with a simple example of a non orthogonal coordinate system.
The coordinate transformation is given by
And its inverse
There are two “natural” ways to specify the coordinate components for the vector in the oblique coordinate system.
Contravariant components =>
The basis vectors are and
Call these covariant basis vectors i = 1..2
and the vector can be written as
= i=1..2 , where theare the vectors contravariant components.
Covariant components =>
The basis vectors are and
Call these contravariant basis vectors i = 1..2
and the vector can be written as
= i=1..2 , where theare the vectors covariant components.
Notice now the joyful fact that
[ ]
The two sets of basis vectors are calledreciprocal.
Using the two reciprocal sets of basis vectors, the scalar product of two vectors simplifies to
Notice that this is now valid in a general curvilinear coordinate system. In Cartesian coordinates the two sets of vector components are equal.
This simple example should clarify why tensor calculus and general curvilinear coordinate systems needs the covariant and contravariant concept(s).
Concepts that are linked to reciprocal basis vectors are dual basis vectors ( and dual vector spaces ( ).
1.2.General curvilinear systems
Generalized coordinates are often denoted by a q and a sub index, like this,, and are a coordinate transformation from a Cartesian orthogonal coordinate system :
= (, , …, i = 1..N [11]
The functions must be differentiable and form an independent set, meaning that the Jacobian determinant must be # 0. The Jacobian is defined as the matrix (Jacobian often means Jacobian determinant):
and is often written as
, or/and some more versions.
and the corresponding determinants with bars around in the usual way or det. If the Jacobian determinant is # 0 the functions can be expressed in :
= (, , …, ) i = 1..N [12]
A line element, displacement vector, in curvilinear coordinates is given by (chain rule):
[13]
The vectorsprovide a natural vector basis and can be visualized starting with a vector and varying one at a time.
The reciprocal vector basis is, the vectors, can be visualized by having fixed and vary the other two. The vector is perpendicular to the surface generated in this way.
vector
Surface:
Denote = and = and that they are reciprocal basis vectors can be easily shown:
[14]
The = basisis particularly appropriate for vectors such as the velocity. The velocity componentsin Cartesian coordinates are simply in the system :
On the other hand, the = basis is appropriate for the gradient operator as, by the chain rule again, the components of the gradient are simply in the basis :
In general, any vector can be expanded in terms of either basis.
The vector components with a sub index are called the covariant components and the ones with a super index are called the contravariant components.
The scalar products = form a second-rank tensor that describes all the angles between the basis vectors and all the lengths of the vectors. It is called the covariant metric tensor and is denoted by . It is obviously symmetric.
[15]
The scalar products= form a second-rank tensor that describes all the angles between the basis vectors and all the lengths of the vectors. It is called the contravariant metric tensor and is denoted by . It is obviously symmetric.
= [16]
The concept of metric tensor is one of the cornerstones of geometry in general curvilinear coordinate systems and differential geometry. The value of the metric tensor is a function of the position in space, i.e. it’s values are generally different at different positions, and
One frequent use is that it can lower and rise the indices of the components of a vector.
and similarly [17]
Example:
[18]
Example:
[19]
2.Tensors
This chapter notes some facts about tensors that are more or less relevant to this article. The important points are marked with a @ symbol.
Einstein’s summation convention is used.
A tensor can be seen as a generalization of the vector concept and is defined by how it is transformed by a coordinate transformation.
The definition of tensors via transformation properties conforms to the physicist’s notion that physical observables must not depend on the choice of coordinate frames.
@ The “main theorem” of tensor calculus is as follows:
If two tensors of the same type are equal in one coordinate system, then they are equal in all coordinate systems.
@ A tensor has a rank.
A scalar has rank 0 and is an “invariant object in the space with respect to the group of coordinate transformations”.
A vector is a rank 1 tensor with covariant components and contravariant components .
Examples of rank 2 tensors are the metric tensor (covariant) (contravariant), the inertia tensor
and a rank 2 mixed tensor (
@ Definition of a rank 1 tensor:
Taking a differential distance vector and letting be a function of the unprimed variables.
[21]
Any set of quantities that transform according to
[22]
is defined as a contravariant vector (contravariant rank-1 tensor), and the indices are written as superscript.
Taking a scalar field the transformation is different.
=>
[23]
Notice the difference, it is vital. [2-3]is the definition of a covariant vector. Rewritten it looks like this
[24]
In the same way tensor rank 2 is defined. When the rank is 2 there is also mixed tensors, one subscript index and one superscript.
@ Definition of a rank 2 tensor:
[25]
[26]
[27]
@ The quotient rule.
To establish the tensor nature of a quantity can be tedious. Help comes from the quotient rule:
If A and B are tensors, and if the expression A = BT is invariant under coordinate transformation, then T is a tensor.
Example:If A and B are tensors and the expression holds in all (rotated) Cartesian coordinate systems, then K is a tensor in the following expressions.
andandand and
@ Contraction
Dealing with vectors in orthogonal coordinates the scalar product is
,with an implicit summation over i
The generalization of this in tensor analysis is a process known as contraction. Two indices, one covariant and the other contravariant, are set equal to each other, and then (as implied by the summation convention) we sum over this repeated index.
The scalar product in a general coordinate system:
3.Christoffel symbols, 1st and 2nd kind
Normal partial derivatives of a vector doesn’t transform as tensors under general curvilinear coordinate transformations. An important property of tensors is that if two tensors A and B are equal, A=B, in one coordinate system then the transformed tensors are equal, A’ = B’.
This property means that two different observers in different coordinate systems agree on physical laws. Substituting regular partial derivatives with the covariant derivatives, which follows tensor transformation rules, is therefore important and has been stated as the mathematical statement of Einstein’s equivalence principle.
The covariant derivative is defined in the next chapter.
The Christoffel symbols have to be defined first.
Starting with scalar
Since the are the components of a contravariant vector, the partial derivatives must form a covariant vector by the quotient rule. The gradient of the scalar becomes
[31]
Moving on to the derivatives of a vector, the situation is more complicated because the basis vectors and are not constant. With vector ,, we get
[32]
or in component form, direct differentiation
[33]
The right hand side of [3-3] differs from the transformation law for a second-rank mixed tensor by the second term containing second derivatives of the coordinates .
will be some linear combination of , write this as
[34]
Multiply by and use to get
[35]
These are Christoffel symbols of the second kind. They are not third-rank tensors. Andisnot generally a second-rank tensor.
, meaning that these Christoffel symbols are symmetric in the lower indices.
[36]
Christoffel symbols of the first kind can be defined as
[37]
The symmetry [ij, k] = [ji, k] follows from second kinds symmetry. [ij, k] is not a third-rank tensor.
. …[38]
3.1.Christoffel symbols as Derivatives of the Metric Tensor
[ definition of covariant metric tensor ]
Differentiate to get:
= [equation [3-8]] = [ik, j] + [jk, i] [39]
Equation [3-9] yields
[ij, k] = [310]
This is the sought expression for Christoffel symbols of the first kind.
Using equation [3-7] :
[311]
Equations [3-10] and [3-11] gives the sought expression for Christoffel symbols of the second kind.
[312]
4.Covariant derivative
Equation [3-2] : can now be rewritten using the Christoffel symbols
and in the last term the k and i indices are dummy indices, change k -> i and i -> k to get
[41]
The expression within the parentheses is the covariant derivative of the contravariant vector and the notation for derivation is a semicolon, not a comma as in chapter about index notation.
[42]
is the covariant derivative of a contravariant vector. It is a second-rank tensor.
By differentiation of the relation it is quite easy to get the expression for the covariant derivative of a covariant vector.
[43]
is the covariant derivative of a covariant vector. It is a second-rank tensor.
A differential becomes
[44]
In Cartesian coordinates the Christoffel symbols vanish and the ordinary partial derivative coincide with the covariant derivative.
A more detailed proof that the covariant derivative is a tensor can be found in [Heinbockel] .
Rules for covariant differentiation:
-The covariant derivative of a sum is the sum of covariant derivatives
-The covariant derivative of a product of tensors is the first times the covariant derivative of the second plus the second times the covariant derivative of the first.
-Higher derivatives are defined as derivatives of derivatives. But take care, in general .
5.The geodesic equation
A geodesic in Euclidean space is a straight line. In general, it is the curve of shortest length between two points and the curve along which a freely falling particle moves. The ellipses of planets are geodesics around the sun, and the moon is in free fall around the Earth on a geodesic.The geodesic can be obtained in a number of ways. We show three of them.
#1The geodesic can be obtained from variational principles, [Arfken] 6th Edition.
[51]
where is the metric of the space.
The variation of
[52]
Inserting [5-2] into [5-1] yields
= 0. [53]
where ds measures the length on the geodesic.
The variations expressed in terms of the independent variations yields
[54]
Insert [5-4] in equation [5-3], shift the derivatives in the last two terms of [5-3] upon integrating by parts and rename the dummy summation indices and [5-3] will be turned into
=0 …[55]
The can have any value, which means that the integrand of [5-5], set equal to zero, gives the geodesic equation. It needs some more manipulations, though …
[56]
Usingand in [5-6] and that is symmetric results in :
= 0 . …[57]
Multiplying [5-7]with and using the fact thatfinally yields the geodesicequation:
= 0 …[58]
The coefficient of the velocities is the Christoffel symbol
#2Analternative derivation of the geodesic equation can be found in [Arfken] 7th Edition.
The distance between two points can be represented as
[59]
To find the geodesic equation it is possible to start from the action:
[510]
Using the Lagrangian formulation of relativistic mechanics where, for a particle not subject to a potential other than a gravitational force (which is described by the metric tensor), the Lagrangian reduces to:
L = [511]
It can be shown that the above Lagrangian leads in fact to the same Euler-Lagrange equations as the Lagrangian relative to [5-9].
We can replace the minimization of J by that of the action:
= 0 [512]
And thereby simplifying the problem by eliminating the radical (the square root).
The minimization in [5-12] is a relatively simple standard problem in variational calculus.
Note that is a function of all the but not on the derivatives . There will be an Euler equation for each k:
= 0 [513]
Evaluate[5-13] to get:
= 0 [514]
Simplify by using:
[515]
And [5-14] can be written as:
[516]
As a final simplification, multiply [5-16] by and use the identity to get the geodesic equation:
[517]
Or using the Christoffel symbols of the second kind written in terms of the metric tensor:
[518]
#3Yet another way to derive the geodesic equation is to see the geodesic as the curve with zero tangent acceleration. The approach can be described as “take a parameterized curve and let the space curvature, described by the metric tensor, move all points along the curve to the correct positions”
Consider a curve
[519]
whichis a sufficiently smooth function and
where
Calling t the ‘time’, only a choice we make, we can call
[520]
the velocity and
[521]
the acceleration.
The geodesic is the shape of the curve when the acceleration has zero projection to the plane tangent to the given surface, which gives the equations
[522]
and using [521] to get
and just do the multiplication
[523]
Before going on, some clarification of the meaning of the condition stated in [5-22] :
It has the form , in vector form ,
which means that gives , the vector has no projection in the direction. Applied to [5-22] this means that .Acceleration has zero projection etc. as stated above. The acceleration, ‘force’, is perpendicular to the curve and only changes the direction of the curve in N-dim space. This condition is then expressed in terms of the metric tensor for the space.
To get the geodesic expressed in terms of the metric tensor, note the definition of the covariant metric tensor
definition of the covariant metric tensor.
Derivation of the definition with respect to yields
[524]
and the two permutations of the indices
[525]
[526]
[5-24] +[5-25] -[5-26]gives :
[527]
Using the definition of the metric tensor + [5-27] into[5-23] :
[528]
Multiplying with and noticing the identity , [528] will result in:
[529]
This is the geodesic equation and it can be written more compact as before using the Christoffel symbol of the second kind:
[530]
An Example“along the geodesic”:
Since the length along the geodesic is a scalar, the velocities of a freely falling particle along the geodesic form a contravariant vector. Hence is a well-defined scalar on a geodesic, which we can differentiate in order to define the covariant derivative of any covariant vector .
The quotient theorem tells us thatis a covariant tensor that defines the covariant derivative of Similarly, higher-order tensors may derived.
Some concluding remarks:
The Mass and Space ‘marriage’, stolen from somewhere, “Mass tells space how to curve and curved space tells mass how to move”.
And as a reminder that there is always more to learn, Einstein’s field equations that are still researched.
This article hopefully gives a hint to understand some of the terms :
where is the Ricci curvature tensor, the scalar curvature, the metric tensor, is the cosmological constant, G is Newton's gravitational constant, c the speed of light in vacuum, and the stress–energy tensor.
6.Appendix A, a wakeup problem
Living in flat Eucledian space ?
The following example serves as an illustration of the importance of the geometry of space itself and the importance of choosing a proper coordinate system.
Here is the problem:
We have a box whose short sides are squares,
1.2 meters x 1.2 meters.
The length of the box is 3 meters.
In the middle of one of the short sides, 0.1 meter from the top side of the box is a spider.
In the middle of the other short side, 0.1 meter from the bottom side of the box, is a fly, caught in the spider’s web.
The spider wants to catch the fly as fast as possible.
The spider can only move on the surface of the box.
The geometry is strictly Euclidean on the surfaces of the box, but with ‘singularities’ at the edges.
How long is the shortest path from the spider to the fly ?