4

Starting GR

Return to Modern Relativity

4.1 The Conceptual Premises For GR

Lets say that there is a space-lab out in the depths of space sealed up so
that there is no way for its crew to see anything outside of the lab. There are
two experimentalists, Terrance and Stella, inside the space-lab. In this
environment they are weightless and Terrance is still with respect to the ship
walls. Stella is also initially still with respect to the lab walls, but she can
maneuver around without touching the walls because she wears a rocket pack. They
both also carry with them cesium watches that keep time accurate to within a
millionth of a second and a computer that can read off such small time
differences in their displays. They then do the following experiment. They
synchronize their watches to start and they start at the same location within
the space-lab. Terrance stays there and Stella travels away and back to him
along any number of paths so long as she arrives back when his watch says an
hour has gone by to within its millionth of a second accuracy. The watch's times
are then compared and a path is sought for which as much time as possible goes
by on Stella's watch. Finally they experimentally discover what we knew from
special relativity which is that the path that maximized her watches time was
simply where she stayed put weightless next to Terrance and didn't go anywhere
else. Every other path she took she underwent special relativistic time dilation
while in motion with respect to Terrance.

Next we shift perspectives to a third party, Lois, who is for the moment
moving in a state of constant velocity through the ship. According to Lois the
path that Stella followed that maximized Stella's time between the events of the
experiment's start and stop next to Terrance was a path of constant velocity. So
we see that in special relativity the paths things tend to take which are paths
of constant velocity are also the paths that maximize proper time intervals
between events along the path.

Next they do another experiment. Lois releases two balls of different mass.
They are both unacted on by forces in the ship so they just keep their same
motion of constant velocity right along with Lois without deviating away from
each-other.

37

38 Chapter 4 Starting GR

Next we go to a fourth observer, Clark Kent, who is far out in the depths of
space, but can see through the walls of the space-lab into the experiments. He
also sees that their space-lab is falling toward a planet which they didn't
realize because they were in free fall and couldn't see outside their lab.
According to Clark the path of maximal proper time that Stella took between the
events of the beginning and the ending of her experiment was not a path of a
constant velocity state at all, but was the path of a body accelerating in the
presence of a gravitational field. So we note that the path that things tend
to follow in gravitational fields are still paths of maximal proper time
even though they are not paths for a constant velocity state.

He also notices that the balls of the experiment though they have different
masses, accelerate at the same rate.

Through this mind experiment we have discovered the core essence of general
relativity.

The equivalence principle comes in different strengths.

The weak version of the equivalence principle boils down to the
equivalence of gravitational and inertial mass. "Gravitational mass" and
"inertial mass" are Newtonian concepts refering to variables that enter into
equations for Newtonian physics. In Newtonian the gravitational force f
from a point active gravitational mass M1 acting on a point passive
gravitational mass M2 at a distance r comes from

fr = -GM1M2/r2

(4.1.1)

In Newtonian physics we also write the relation between the fr
acting on an inertial mass M2i and ar as

fr = M2iar

(4.1.2)

Putting these together we have

ar =
(-GM1/r2)(M2/M2i)

(4.1.3)

We noted the balls of different masses fell at the same rate of acceleration
according to Clark. In order for this acceleration to be independent of the ball
mass as Clark saw that it was, with the correct choice for the value of G it
becomes clear that the gravitational mass M2 must be equivalent to
inertial mass M2i. Then we have

ar = -GM1/r2

(4.1.4)

In general relativity we will have an invariant definition of mass just as we
have defined mass as invariant in the special relativity chapters. There will
also be a four-vector force equation for general relativity in the form

Fl = mAl

where m is the mass as invariant for general relativity.

Gravitation acting alone corresponds to Fl = 0. This yields:

mAl = 0

The Acceleration four-vector for general relativity is a combonation of two
parts discussed in more detail later resulting in

mdUl/dt + mGlmnUmUn = 0

The m in the term on the left corresponds to the "inertial mass" in Newtonian
physics. The m in the term on the right corresponds to the passive
"gravitational mass" in Newtonian physics. As these are really the same thing
that was just multiplied through it is obvious that indeed the inertial and
gravitational masses are identically equivalent.

4.1 The Conceptual Premises For GR 39

The semi-strong level of the equivalence principle comes from
the realization that the crew never knew that they were actually falling in a
gravitational field. The experiments of a local free fall frame have results
indistinguishable from the same experiments done in inertial frames. This is an
equivalence of inertial and local free fall frames. We could also extend
this to the realization that if the lab had rocket engines burning, keeping them
at a constant proper acceleration, they wouldn't have known the difference
between being accelerated by the rocket engines or sitting on the surface of a
planet in the presence of a gravitational field.

The strong level of the equivalence principle comes from the
realization that any local free fall frames are equivalent for doing the
physics. The laws of physics were the same for Lois as they were for Terrance.
When the equivalence principle is mention unqualified it is usually this level
of equivalence that is being referred to.

Above this strength we find the level of equivalence that is really required
to result in the form of general relativity that we have today. This is
sometimes called the general principle of relativity and sometimes the
general principle of covariance. That is simply the statement that the
general laws of physics are frame covariant. In other words the equation form
that the laws of physics take are the same, invariant, according to every frame
whether accelerated or not, whether in the presence of a gravitational field or
not, whether rotating or not. To ensure this we must model the general laws of
physics with tensor equations. The equations for the general laws of physics are
then unchanged by transformations.

Exercises

Problem 4.1.1

If a laser is mounted on the bottom of an elevator in free fall, would a
passenger notice any red shift?

Problem 4.1.2

If a laser were mounted on the side of an elevator in free fall, would a
passenger notice any bend in the beam? What about an observer standing on the
ground outside?

Problem 4.1.3

If a spaceship orients a laser in the direction of its acceleration, do the
passengers observe a red shift? What does the result mean for the rate clocks
high in the ship run compared to clocks low in the ship? What would this mean
for clocks hovering at different heights over a planet?

_______________________________________________________________________________________

40 Chapter 4 Starting GR

4.2 Tensors in GR

What defines a vector in any physics is its vector transformation properties.
Not everything that merely has a magnitude and a direction is a vector, even in
non-relativistic physics. For instance angular displacement is not really
a vector because it doesn't always obey the vector property

A + B = B + A.

The vectors of relativity obey tensor transformation properties. In
general, a four-vector is a rank one tensor. In element notation is has
only one index, so it is a tensor with only four elements.

Some of the things we like to think of as individual properties of nature are
incomplete as physical properties being only a component of a tensor. For
instance, the electric field by itself does not obey tensor transformation
properties. The magnetic field by itself also does not obey tensor
transformation properties. In the context of this text a pseudovector will be
anything that has multiple elements like a vector, but lacks any of the tensor
transformation properties. These two pseudo-vectors can be combined into a
unified field called the electromagnetic field tensor. Thus we see that
the electric and magnetic fields are actually incomplete parts of the actual
unified field called the electromagnetic field. This is a rank two tensor.

In the same sense, momentum by itself is not a complete physical quantity as
it does not obey tensor transformation properties and so it is not really a
vector in the relativistic sense. But, when we combine it with a fourth element,
energy, we get a tensor called the momentum four-vector.

Likewise there are displacement four-vectors, velocity four-vectors,
acceleration four-vectors, force four-vectors, etc...

According to a general principle of relativity the laws of physics are frame
covariant. Therefor when modeling the general laws of physics with equations we
must use expressions that are also frame covariant. For instance, if we use one
coordinate system to write an equation like

F(ct,x,y,z) - G(ct,x,y,z) = 0,

Then in any other coordinate system it should also be

F ' (ct',
x',y',z' ) - G '
(ct', x',y',z' ) = 0

It should not change its basic form. For example, it should not become

F ' (ct',
x',y',z' ) - G '
(ct', x',y',z' ) = H
' (ct', x',y',z'
)

If such an equation does transforms like this then it is not one of the
fundamental equations of physics.

4.2 Tensors in GR 41

Here we will define a tensor in terms of its transformation properties. A
contravariant tensor will be any quantity that transforms between frames
according to

T ' m =
(¶x' m
/¶xn )T
n

(4.2.1)

A covariant tensor will be any quantity that transforms between frames
according to

T ' m = (¶x n /¶x' m )T n

(4.2.2)

There are also mixed tensors. For example

T ' m
n = (¶x' m /¶xs )(¶x r /¶x' n )T
s r

(4.2.3)

From these transformation properties we can deduce that for an individual
particle,

1.) A sum or difference of tensors is still a tensor.

2.) A product of tensors is still a tensor.

3.) A tensor multiplied or divided by an invariant is still a tensor.

[note - these rules apply only when the tensors involved describe that which
is observed, not the state of the observer himself. So for example let
Fmn be a tensor describing something
observed like say the electromagnetic field and Un is the four-vector velocity of the observer
(c,0,0,0). It turns out that the electric field given by

Em = Fm0 = FmnUn/c

is NOT a tensor. As Un is the four-vector
velocity of whoever is the observer everyone uses (c,0,0,0) as a result and the
expression does not transform as a four-vector. E'm = F'm0
¹ (¶xl/¶x'm)Fl0. If
Un were the four-vector velocity of one
"particular" observer then the expression would transform as a tensor, but then
it wouldn't represent the electric field to anyone except that observer and it
would then only when Fmn is the
electromagnetic field already expressed according to his own frame. Likewise the
magnetic field

Bm = - (1/2)em0lrFlr/c = -
(1/2)emnlrFlrUn/c2

where Un is the four-velocity of the
observer (c,0,0,0) is also not a tensor.]

In relativity we must write the fundamental equations of physics as tensor
equations such as

T ms ...nr ... =
0

(4.2.4)

because this remains frame covariant. For instance, using the above
transformation properties, it is easy to show that in any other frame this
equation remains in the same form

T ' ms ...nr
... = 0

Exercises

Problem 4.2.1

Use the general relativistic definition of a tensor to show that for an
individual particle,

1.) A sum or difference of tensors is still a tensor.

2.) A product of tensors is still a tensor.

3.) A tensor multiplied or divided by an invariant is still a tensor.

[note - these rules apply only when the tensors involved describe that which
is observed, not the state of the observer himself. So for example let
Fmn be a tensor describing something
observed like say the electromagnetic field and Un is the four-vector velocity of the observer
(c,0,0,0). It turns out that the electric field given by

Em = Fm0 = FmnUn/c

is NOT a tensor. As Un is the four-vector
velocity of whoever is the observer everyone uses (c,0,0,0) as a result and the
expression does not transform as a four-vector. E'm = F'm0
¹ (¶xl/¶x'm)Fl0. If
Un were the four-vector velocity of one
"particular" observer then the expression would transform as a tensor, but then
it wouldn't represent the electric field to anyone except that observer and it
would then only when Fmn is the
electromagnetic field already expressed according to his own frame. Likewise the
magnetic field

Bm = - (1/2)em0lrFlr/c = -
(1/2)emnlrFlrUn/c2

where Un is the four-velocity of the
observer (c,0,0,0) is also not a tensor.]

_______________________________________________________________________________________

42 Chapter 4 Starting GR

4.3 The Metric and Invariants of GR

Recall that for special relativity the invariant interval can be
expressed in the form Eqn 2.2.3

ds2 = dct2 - dx2 -
dy2 - dz2

Or in a more compact notation it can be written Eqn 2.2.5

ds2 = hmndxmdxn

If we were to express this in a curvilinear coordinate system it will take on
a form different from the top equation. For example do the following
transformation to cylindrical coordinates

x = rcosq

y = rsinq

The invariant interval will then take the form

ds2 = dct2 - dr2 -
r2dq2 - dz2

Notice that in curvilinear coordinate systems functions of the coordinates
may appear as coefficients of the differential quantities within the interval
such as the

-r2 appears front of the dq2
term above. Another possibility is the appearance of cross terms such as a dctdz
term. To write this as a more compact and general form it is expressed

ds2 = gmndxmdxn

(4.3.1)

When there is matter or fields of any type in the space it effects the form
that gmn can take globally. So the popular
interpretation for gravitation is simply that matter gives space-time an
intrinsic curvature. In a situation where matter curves the space-time one can
not globally transform gmn to hmn. However one can always do the transformation
locally.

We again express the invariant interval in the form

ds2 = gmndxmdxn

Given that the interval is invariant we know that

gmndxmdxn =
g'lrdx'ldx'r

We also know that dxm transforms
according to the calculus chain rule

dx'm = (¶x'm/¶xn)dxn

4.3 The Metric and Invariants of GR 43

This results in

gmndxmdxn = (¶x'l/¶xm)(¶x'r/¶xn)g'lrdxmdxn

And therefor

gmn = (¶x'l/¶xm)(¶x'r/¶xn)g'lr

Now this is how a rank 2 covariant tensor transforms. Therefor if
ds2 is to be invariant then gmn
is a rank 2 covariant tensor. This has been given the name "the metric
tensor"

As we shall cover in the sections on gravitational pseudo forces the metric
tensor is analogous to the gravitational potential for non-relativistic physics.
In non-relativistic physics the gravitational force or other fields are often
describable as the gradient of a potential. In later sections the gravitational
pseudo forces will be related to affine connections which contain the metric
tensor and its first order derivatives.

For special relativity we have

hmn,n= 0

We can always transform to a local frame according to which the metric is
hmn so we know so far that for a local frame
also

hmn,n= 0

Now consider the transformation to be to a local free fall frame so that the
affine connections vanish. In that case we also have

hmn;n= 0

Now transform this result to an arbitrary frame and we also find

gmn;n = 0

(4.3.2)

(Summation still implied on all four above)

Next consider the quantity

gmrgrn

as arrived at for any point in spacetime by a transformation to an arbitrary
set of Coordinates from a local Cartesian coordinate frame:

gmrgrn = (¶xa/¶x'm)(¶xb/¶x'r)hab(¶x'r/¶xl)(¶x'n/¶xs)hls

Rearrange terms

gmrgrn = (¶xa/¶x'm)(¶xb/¶x'r)(¶x'r/¶xl)(¶x'n/¶xs)habhls

Yielding

gmrgrn = (¶xa/¶x'm)dbl(¶x'n/¶xs)habhls

Simplify

gmrgrn = (¶xa/¶x'm)(¶x'n/¶xs)habhbs

From the matrix equation for hmn it is easy to verify the next step

gmrgrn = (¶xa/¶x'm)(¶x'n/¶xs)das

Simplify

gmrgrn = (¶xa/¶x'm)(¶x'n/¶xa)

44 Chapter 4 Starting GR

This yields

gmrgrn = dmn

(4.3.3)

Contract this and we have

gmrgrm = dmm

Which results in

gmngmn = 4

(4.3.4)

The covariant metric tensor also acts as a lowering index operator and the
contravariant metric tensor acts as a raising index operator. For example,

Tm = gmnTn

and (4.3.5)

Tm = gmn Tn

It is easy to verify this property based how contravariant and covariant
tensors are defined by how they transform. For example consider the following
expression,

(¶x l
/¶x' m
)(glnTn)

based on how tensors transform this becomes

(¶x l
/¶x' m
)(¶x' a
/¶xl )(¶x' b /¶xn )g'ab(¶x n /¶x' r )T ' r = (¶x l /¶x' m )(glnTn)

Rearranging:

(¶x l
/¶x' m
)(¶x' a
/¶xl )(¶x' b /¶xn )(¶x n /¶x' r
)g'abT '
r = (¶xl /¶x' m )(glnTn)

Recognizing these result in delta Kroneckers and collecting the priming it
becomes,

dmadbr(gabTr)' =
(¶x l /¶x' m
)(glnTn)

This simplifies to

(gmrTr)' = (¶x l /¶x' m )(glnTn)

But then we recognize that this is how a covariant tensor transforms and so
we name Tm by calling it,

Tm = gmnTn

4.3 The Metric and Invariants of GR 45

Thus we've verified the lowering index property of the covariant metric
tensor. Verifying the raising index property of the contravariant metric tensor
is easier at this point. Start with the expression,

gmnTn

We've named our previous expression Tn
and so we insert it.

gmngnrTr = gmnTn

But we've already verified that gmngnr = dmr so we have

dmrTr = gmnTn

Which results in

Tm = gmnTn

This verifies the raising of index property of the contravariant metric
tensor.

With the exception of the locations of physical singularities, the space-time
for the universe in which we live is an everywhere locally Lorentzian spacetime.
A locally Lorentzian spacetime is a spacetime for which we can locally transform
gmn to hmn where hmn
is given by Eqn 2.2.4

A locally Euclidean Space-time is a spacetime for which we can locally
transform gmn to wmn where wmn is
given by

(4.3.6)

In other words all the dimensions of a Euclidean "spacetime" are
spacelike.

Either type of spacetime can have Riemannian Curvature as these are only
locally Euclidean, or Lorentzian.

46 Chapter 4 Starting GR

Note- Sometimes it is said that our Universe is everywhere locally Euclidean.
This basically means that we can do local transformations to arrive at

(4.3.7)

This is correct, but to prevent confusion it is really more appropriate to
say that our universe is everywhere locally Lorentzian.

Our universe is also described as being a globally Riemannian spacetime. This
means that it globally takes the quadradic form of Eqn. 4.3.1

ds2 = gmndxmdxn

and is the same thing as saying it is everywhere locally Lorentzian.

An invariant as defined for this text is a quantity whose value does not
depend on speed, location with respect to gravitational sources etc... nor upon
whose frame it was calculated from. Invariants are said to be invariant to frame
transformations, or frame invariant. This does not imply that the value of an
invariant must be the same everywhere (for example invariant "densities") nor
that it must be conserved. In this context an invariant can be thought of as
short for invariant scalar though there are tensor expressions such as the delta
kronecker tensor whose elements are all frame invariant. Some people also think
of tensors in general as invariants as they represent physical entities and
physical entities will not depend in any intrinsic way on our choice of frame.
From this perspective the "elements of a tensor" are thought of as "projections
of the tensor" onto a coordinate dependent template. The paradigm for this text
will instead be that the tensor is the template onto which the projections have
been made. It is not invariant, but transforms according to the transformation
properties of an infinitesimal displacement vector. Some relativity authors use
the word scalar to be short for invariant scalars or what are just called
invariants in this text. This is popular, but extremely inappropriate. The
reason that it is inappropriate is that if people continue to redefine things
without good reason so that they have a different meaning for whatever theory
comes along then when they are used in general, eventually a student will
practically have to learn a different dialect of the spoken language for every
theory encountered. This is complication beyond reason. Here are a few examples
of invariants

c The local vacuum speed of light

m Mass

p The pressure scalar [p = (1/3)(TmnUmUnc -2 - gmnTmn), for
example the pressure of a gass]

t The proper time between events along a world line.

q Charge

An example of how one of these invariants might not be conserved would be to
consider the pressure of the gas after a balloon is popped in space. As it
expands the pressure decreases and so it is not conserved.

An example from special relativity of a quantity that is conserved, but
not invariant would be the total energy of a particle E.

An example of a quality that is both invariant and conserved would be total
charge q.

4.3 The Metric and Invariants of GR 47

Consider the transformation of the full contraction of a tensor Tm.

g'mnT
'mT
'n = [(¶xl/¶x'm)(¶xr/¶x'n)glr][(¶x'm/¶xa)Ta][(¶x'n/¶xb)Tb]

g'mnT
'mT
'n = (¶xl/¶x'm)(¶x'm/¶xa)(¶xr/¶x'n)(¶x'n/¶xb)glrTaTb

g'mnT
'mT
'n = dladrbglrTaTb

g'mnT
'mT
'n= glrTlTr

So we note that the full contraction of a tensor is an invariant.

Exercises

Problem 4.3.1

Find hlrPlPr. If the
contraction of a tensor is an invariant and this was a local result for the
contraction of Pm, what does this tell you
about mass in general relativity?

Problem 4.3.2

Write out Pm for gmn = hmn .

Problem 4.3.3

Recall problem 2.2.3. What does dt/dt' turn out to be for the
spacetime

ds2 = (1 + az/c2)2dct2 - dx2
- dy2 - dz2

Problem 4.3.4

Use Eqn 4.3.3 to find gmn for the
spacetime in Problem 4.3.3. Hint - this is simply a matrix inversion.

Problem 4.3.5

Write out gmn for

ds2 = (A2 + gzzb2(1 - f)2)dct2 +
2gzzb(1 - f)dctdz +
gzzdz2 - H2(dx2 +
dy2)

Hint - gmn is symmetric and gmn is a matrix inversion of gmn.

Problem 4.3.6

Consider the spacetime

ds2 = (1 - r0/r)dct2 -
dr2/(1 - r0/r) - r2(dq2 + sin2qdf2)

Write out Tm for an arbitrary vector
Tm and find TmTm.

_______________________________________________________________________________________

48 Chapter 4 Starting GR

4.4 The Affine Connections and The Covariant
Derivative

We want to make equations for the general laws of physics out of tensor
equations. So in developing a differentiation operator for general relativity we
must assure that when it is operated on a tensor it results in something that is
still a tensor. We find that many of the special relativistic laws of physics
are described by equations involving ordinary differentiation and so this
operator must also reduce to the ordinary differentiation operator in local free
fall frames. Consider the chain rule for the ordinary differentiation of a
tensor.

dTl = (¶Tl/¶xr)dxr

Using the transformation property of a contravariant tensor we find

dTl = {(¶/¶xr)[(¶xl/¶x '
s)T ' s]}dxr

Using the product rule we come to

dTl = (¶2xl/¶xr¶x
' s)T'sdxr + (¶xl/¶x ' s)(¶T '
s/¶xr)dxr

And again from the chain rule we finally have

dTl = (¶2xl/¶xr¶x
' s)T '
sdxr + (¶xl/¶x ' s)dT ' s

Now if on the right hand side we only had the second term then the
differentiation of a tensor would still transform as a tensor, but we have the
extra first term so we know it does not. Thus to find a differentiation operator
which maps tensors to tensors we introduce a second term in the operation. The
new differential operator is called the covariant derivative
opperator.

DTl = dTl + dTl

(4.4.1)

For a contravariant vector the second term necessary to keep DTl a tensor is

dTl = GlmnTmdxn

(4.4.2)

where the affine connection(sometimes called the Christophel symbol of
the second kind) Glmn is given
by

Glmn = (1/2)glr(gmr,n +
gnr,m - gmn,r)

(4.4.3)

4.4 The Affine Connections and The Covariant Derivative 49

For covariant four vectors we can write it in the same form

DTl = dTl + dTl

(4.4.4)

But here we have

dTl = - GmlnTmdxn

(4.4.5)

In the case of the differentiation of a multiple mixed rank tensor we find

DTl...k ... = dTl...k
... + GlmnTm...k
...dxn +... - GrknTl...r
...dxn -...

(4.4.6)

Also it is important to make note that though the affine connection is a part
of a covariant derivative operator, it is not a tensor itself.

So, for example, the covariant derivative of a tensor Tl with respect to some invariant parameter such as
dt is

DTl/dt = dTl/dt + GlmnTm(dxn/dt)

(4.4.7)

As mentioned, a comma will represent a partial derivative and a semicolon
will represent a partial covariant derivative. So for example

Tl;r = Tl,r + GlmnTm(¶xn/¶xr)

This simplifies to (4.4.8)

Tl;r = Tl,r + GlmrTm

Exercises

Problem 4.4.1

Work out the affine connections and verify Eqn 4.3.2 for the spacetime in
problem 4.3.3.

Problem 4.4.2

Given that the metric tensor is symmetric, verify that the Affine connections
are symmetric in the lower indices. That is to say verify Glmr = Glrm.