equivalence Modern Relativity modernrelativity = special general black hole mass energy Einstein wormhole time = Schwarzschild light

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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=20 that there is no way for its crew to see anything outside of the lab. = There are=20 two experimentalists, Terrance and Stella, inside the space-lab. In this = environment they are weightless and Terrance is still with respect to = the ship=20 walls. Stella is also initially still with respect to the lab walls, but = she can=20 maneuver around without touching the walls because she wears a rocket = pack. They=20 both also carry with them cesium watches that keep time accurate to = within a=20 millionth of a second and a computer that can read off such small time=20 differences in their displays. They then do the following experiment. = They=20 synchronize their watches to start and they start at the same location = within=20 the space-lab. Terrance stays there and Stella travels away and back to = him=20 along any number of paths so long as she arrives back when his watch = says an=20 hour has gone by to within its millionth of a second accuracy. The = watch's times=20 are then compared and a path is sought for which as much time as = possible goes=20 by on Stella's watch. Finally they experimentally discover what we knew = from=20 special relativity which is that the path that maximized her watches = time was=20 simply where she stayed put weightless next to Terrance and didn't go = anywhere=20 else. Every other path she took she underwent special relativistic time = dilation=20 while in motion with respect to Terrance.

Next we shift perspectives to a third party, Lois, who is for the = moment=20 moving in a state of constant velocity through the ship. According to = Lois the=20 path that Stella followed that maximized Stella's time between the = events of the=20 experiment's start and stop next to Terrance was a path of constant = velocity. So=20 we see that in special relativity the paths things tend to take which = are paths=20 of constant velocity are also the paths that maximize proper time = intervals=20 between events along the path.

Next they do another experiment. Lois releases two balls of different = mass.=20 They are both unacted on by forces in the ship so they just keep their = same=20 motion of constant velocity right along with Lois without deviating away = from=20 each-other.

38 Chapter 4 Starting GR

Next we go to a fourth observer, Clark Kent, who is far out in the = depths of=20 space, but can see through the walls of the space-lab into the = experiments. He=20 also sees that their space-lab is falling toward a planet which they = didn't=20 realize because they were in free fall and couldn't see outside their = lab.=20 According to Clark the path of maximal proper time that Stella took = between the=20 events of the beginning and the ending of her experiment was not a path = of a=20 constant velocity state at all, but was the path of a body accelerating = in the=20 presence of a gravitational field. So we note that the path that = things tend=20 to follow in gravitational fields are still paths of maximal proper = time=20 even though they are not paths for a constant velocity state.

He also notices that the balls of the experiment though they have = different=20 masses, accelerate at the same rate.

Through this mind experiment we have discovered the core essence of = general=20 relativity.

The equivalence principle comes in different strengths.

The weak version of the equivalence principle boils = down to the=20 equivalence of gravitational and inertial mass. "Gravitational = mass" and=20 "inertial mass" are Newtonian concepts refering to variables that enter = into=20 equations for Newtonian physics. In Newtonian the gravitational force = f=20 from a point active gravitational mass M₁ acting on a point = passive=20 gravitational mass M₂ at a distance r comes from

f_r =3D = -GM₁M₂/r²

(4.1.1)

In Newtonian physics we also write the relation between the = f_r=20 acting on an inertial mass M_2i and a_r as

f_r =3D M_2ia_r

(4.1.2)

Putting these together we have

a_r =3D=20 (-GM₁/r²)(M₂/M_2i)

(4.1.3)

We noted the balls of different masses fell at the same rate of = acceleration=20 according to Clark. In order for this acceleration to be independent of = the ball=20 mass as Clark saw that it was, with the correct choice for the value of = G it=20 becomes clear that the gravitational mass M₂ must be = equivalent to=20 inertial mass M_2i. Then we have

a_r =3D -GM₁/r²

(4.1.4)

In general relativity we will have an invariant definition of mass = just as we=20 have defined mass as invariant in the special relativity chapters. There = will=20 also be a four-vector force equation for general relativity in the = form

F^l =3D mA^l

where m is the mass as invariant for general relativity.

Gravitation acting alone corresponds to F^l =3D 0. This yields:

mA^l =3D 0

The Acceleration four-vector for general relativity is a combonation = of two=20 parts discussed in more detail later resulting in

mdU^l/dt + mG^l_mnU^mUⁿ =3D 0

The m in the term on the left corresponds to the "inertial mass" in = Newtonian=20 physics. The m in the term on the right corresponds to the passive=20 "gravitational mass" in Newtonian physics. As these are really the same = thing=20 that was just multiplied through it is obvious that indeed the inertial = and=20 gravitational masses are identically equivalent.

4.1 The Conceptual Premises For GR 39

The semi-strong level of the equivalence principle = comes from=20 the realization that the crew never knew that they were actually falling = in a=20 gravitational field. The experiments of a local free fall frame have = results=20 indistinguishable from the same experiments done in inertial frames. = This is an=20 equivalence of inertial and local free fall frames. We could also = extend=20 this to the realization that if the lab had rocket engines burning, = keeping them=20 at a constant proper acceleration, they wouldn't have known the = difference=20 between being accelerated by the rocket engines or sitting on the = surface of a=20 planet in the presence of a gravitational field.

The strong level of the equivalence principle comes = from the=20 realization that any local free fall frames are equivalent for = doing the=20 physics. The laws of physics were the same for Lois as they were for = Terrance.=20 When the equivalence principle is mention unqualified it is usually this = level=20 of equivalence that is being referred to.

Above this strength we find the level of equivalence that is really = required=20 to result in the form of general relativity that we have today. This is=20 sometimes called the general principle of relativity and = sometimes the=20 general principle of covariance. That is simply the statement that = the=20 general laws of physics are frame covariant. In other words the equation = form=20 that the laws of physics take are the same, invariant, according to = every frame=20 whether accelerated or not, whether in the presence of a gravitational = field or=20 not, whether rotating or not. To ensure this we must model the general = laws of=20 physics with tensor equations. The equations for the general laws of = physics are=20 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=20 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=20 passenger notice any bend in the beam? What about an observer standing = on the=20 ground outside?

Problem 4.1.3

If a spaceship orients a laser in the direction of its acceleration, = do the=20 passengers observe a red shift? What does the result mean for the rate = clocks=20 high in the ship run compared to clocks low in the ship? What would this = mean=20 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.=20 Not everything that merely has a magnitude and a direction is a vector, = even in=20 non-relativistic physics. For instance angular displacement is = not really=20 a vector because it doesn't always obey the vector property

A + B =3D B + A.

The vectors of relativity obey tensor transformation properties. In=20 general, a four-vector is a rank one tensor. In element notation = is has=20 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=20 incomplete as physical properties being only a component of a tensor. = For=20 instance, the electric field by itself does not obey tensor = transformation=20 properties. The magnetic field by itself also does not obey tensor=20 transformation properties. In the context of this text a pseudovector = will be=20 anything that has multiple elements like a vector, but lacks any of the = tensor=20 transformation properties. These two pseudo-vectors can be combined into = a=20 unified field called the electromagnetic field tensor. Thus we = see that=20 the electric and magnetic fields are actually incomplete parts of the = actual=20 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=20 it does not obey tensor transformation properties and so it is not = really a=20 vector in the relativistic sense. But, when we combine it with a fourth = element,=20 energy, we get a tensor called the momentum four-vector.

Likewise there are displacement four-vectors, velocity four-vectors,=20 acceleration four-vectors, force four-vectors, etc...

According to a general principle of relativity the laws of physics = are frame=20 covariant. Therefor when modeling the general laws of physics with = equations we=20 must use expressions that are also frame covariant. For instance, if we = use one=20 coordinate system to write an equation like

F(ct,x,y,z) - G(ct,x,y,z) =3D 0,

Then in any other coordinate system it should also be

F'(ct',=20 x',y',z') - G'⁼²⁰(ct', x',y',z') =3D 0

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

F'(ct',=20 x',y',z') - G'⁼²⁰(ct', x',y',z') =3D H⁼'(ct', = x',y',z'⁼²⁰)

If such an equation does transforms like this then it is not one of = the=20 fundamental equations of physics.

4.2 Tensors in GR 41

Here we will define a tensor in terms of its transformation = properties. A=20 contravariant tensor will be any quantity that transforms between frames = according to

T'^m =3D=20 (=B6x'^m=20/=B6xⁿ )T^=20
n

(4.2.1)

A covariant tensor will be any quantity that transforms between = frames=20 according to

T'_m =3D (=B6x^=
n/=B6x'^m)T_{n =}

(4.2.2)

There are also mixed tensors. For example

T'^m=20_n =3D (=B6x'^{m =}/=B6x^s )(=B6x^r/=B6x'^{n =})T^=20
s_r⁼²⁰

(4.2.3)

From these transformation properties we can deduce that for an = individual=20 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=20 is observed, not the state of the observer himself. So for example let=20 F_mn be a tensor describing = something=20 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

E_m =3D F_m0 =3D F_mnUⁿ/c

is NOT a tensor. As Uⁿ is the = four-vector=20 velocity of whoever is the observer everyone uses (c,0,0,0) as a result = and the=20 expression does not transform as a four-vector. E'_m =3D F'_m0=20 =B9 (=B6x^l/=B6x'^m)F_l0. = If=20 Uⁿ were the four-vector = velocity of one=20 "particular" observer then the expression would transform as a tensor, = but then=20 it wouldn't represent the electric field to anyone except that observer = and it=20 would then only when F_mn is the=20 electromagnetic field already expressed according to his own frame. = Likewise the=20 magnetic field

B_m =3D - (1/2)e_m0^lrF_lr/c = =3D -=20 (1/2)e_mn^lrF_lrUⁿ/c²

where Uⁿ is the four-velocity = of the=20 observer (c,0,0,0) is also not a tensor.]

In relativity we must write the fundamental equations of physics as = tensor=20 equations such as

T^{ms ...}_{nr =
...} =3D=20 0

(4.2.4)

because this remains frame covariant. For instance, using the above=20 transformation properties, it is easy to show that in any other frame = this=20 equation remains in the same form

T'^{ms =
...}_nr=20
... =3D 0

Exercises

Problem 4.2.1

Use the general relativistic definition of a tensor to show that for = an=20 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.

E_m =3D F_m0 =3D F_mnUⁿ/c

B_m =3D - (1/2)e_m0^lrF_lr/c = =3D -=20 (1/2)e_mn^lrF_lrUⁿ/c²

where Uⁿ is the four-velocity = of the=20 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=20 expressed in the form Eqn 2.2.3

ds² =3D dct² - dx² -=20 dy² - dz²

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

ds² =3D h_mndx^mdxⁿ

If we were to express this in a curvilinear coordinate system it will = take on=20 a form different from the top equation. For example do the following=20 transformation to cylindrical coordinates

x =3D rcosq

y =3D rsinq

The invariant interval will then take the form

ds² =3D dct² - dr² -=20 r²dq² - = dz²

Notice that in curvilinear coordinate systems functions of the = coordinates=20 may appear as coefficients of the differential quantities within the = interval=20 such as the

-r² appears front of the dq²=20 term above. Another possibility is the appearance of cross terms such as = a dctdz=20 term. To write this as a more compact and general form it is = expressed

ds² =3D g_mndx^mdxⁿ

(4.3.1)

When there is matter or fields of any type in the space it effects = the form=20 that g_mn can take globally. So the = popular=20 interpretation for gravitation is simply that matter gives space-time an = intrinsic curvature. In a situation where matter curves the space-time = one can=20 not globally transform g_mn to = h_mn. However one can always do the = transformation=20 locally.

We again express the invariant interval in the form

ds² =3D g_mndx^mdxⁿ

Given that the interval is invariant we know that

g_mndx^mdxⁿ = =3D=20 g'_lrdx'^ldx'^r

We also know that dx^m = transforms=20 according to the calculus chain rule

dx'^m =3D = (=B6x'^m/=B6xⁿ)dxⁿ

4.3 The Metric and Invariants of GR 43

This results in

g_mndx^mdxⁿ = =3D (=B6x'^l/=B6x^m)(=B6x'^r/=B6xⁿ)g'_lrdx^mdxⁿ

And therefor

g_mn =3D (=B6x'^l/=B6x^m)(=B6x'^r/=B6xⁿ)g'_lr

Now this is how a rank 2 covariant tensor transforms. Therefor if=20 ds² is to be invariant then g_mn=20 is a rank 2 covariant tensor. This has been given the name "the metric=20 tensor"

As we shall cover in the sections on gravitational pseudo forces the = metric=20 tensor is analogous to the gravitational potential for non-relativistic = physics.=20 In non-relativistic physics the gravitational force or other fields are = often=20 describable as the gradient of a potential. In later sections the = gravitational=20 pseudo forces will be related to affine connections which contain the = metric=20 tensor and its first order derivatives.

For special relativity we have

h_mn,_n=3D 0

We can always transform to a local frame according to which the = metric is=20 h_mn so we know so far that for a = local frame=20 also

h_mn,_n=3D 0

Now consider the transformation to be to a local free fall frame so = that the=20 affine connections vanish. In that case we also have

h_mn;_n=3D 0

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

g_mn;_n =3D 0

(4.3.2)

(Summation still implied on all four above)

Next consider the quantity

g_mrg^rn

as arrived at for any point in spacetime by a transformation to an = arbitrary=20 set of Coordinates from a local Cartesian coordinate frame:

g_mrg^rn =3D (=B6x^a/=B6x'^m)(=B6x^b/=B6x'^r)h_ab(=B6x'^r/=B6x^l)(=B6x'ⁿ/=B6x^s)h^ls

Rearrange terms

g_mrg^rn =3D (=B6x^a/=B6x'^m)(=B6x^b/=B6x'^r)(=B6x'^r/=B6x^l)(=B6x'ⁿ/=B6x^s)h_abh^ls

Yielding

g_mrg^rn =3D (=B6x^a/=B6x'^m)d^b_l(=B6x'ⁿ/=B6x^s)h_abh^ls

Simplify

g_mrg^rn =3D (=B6x^a/=B6x'^m)(=B6x'ⁿ/=B6x^s)h_abh^bs

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

g_mrg^rn =3D (=B6x^a/=B6x'^m)(=B6x'ⁿ/=B6x^s)d_a^s

Simplify

g_mrg^rn =3D (=B6x^a/=B6x'^m)(=B6x'ⁿ/=B6x^a)

44 Chapter 4 Starting GR

This yields

g_mrg^rn =3D d_mⁿ=20

(4.3.3)

Contract this and we have

g_mrg^rm =3D d_m^m=20

Which results in

g_mng^mn =3D 4

(4.3.4)

The covariant metric tensor also acts as a lowering index operator = and the=20 contravariant metric tensor acts as a raising index operator. For = example,

T_m⁼=3D g_mnTⁿ =

and (4.3.5)

T^m =3D = g^mn T_n =

It is easy to verify this property based how contravariant and = covariant=20 tensors are defined by how they transform. For example consider the = following=20 expression,

(=B6x^l=20/=B6x'^m=20)(g_lnTⁿ)

based on how tensors transform this becomes

(=B6x^l=20/=B6x'^m=20)(=B6x'^a=20/=B6x^l )(=B6x'^{b =}/=B6xⁿ = )g'_ab(=B6xⁿ/=B6x'^r)T'^r =3D (=B6x^=
l/=B6x'^m)(g_lnTⁿ)

Rearranging:

(=B6x^l=20/=B6x'^m=20)(=B6x'^a=20/=B6x^l )(=B6x'^{b =}/=B6xⁿ )(=B6xⁿ/=B6x'^r=20)g'_abT⁼'⁼²⁰r =3D (=B6x^l/=B6x'^m)(g_lnTⁿ)

Recognizing these result in delta Kroneckers and collecting the = priming it=20 becomes,

d_m^ad^b_r(= g_abT^r)' =3D=20 (=B6x^{l =}/=B6x'^m=20)(g_lnTⁿ)

This simplifies to

(g_mrT^r)' =3D (=B6x^l/=B6x'^m)(g_lnTⁿ)

But then we recognize that this is how a covariant tensor transforms = and so=20 we name T_m by calling it,

T_m =3D = g_mnTⁿ

4.3 The Metric and Invariants of GR 45

Thus we've verified the lowering index property of the covariant = metric=20 tensor. Verifying the raising index property of the contravariant metric = tensor=20 is easier at this point. Start with the expression,

g^mnT_n

We've named our previous expression T_n=20 and so we insert it.

g^mng_nrT^r = =3D g^mnT_{n =}

But we've already verified that g^mng_nr = =3D d^m_r so we have

d^m_rT^r =3D g^mnT_n

Which results in

T^m =3D = g^mnT_{n =}

This verifies the raising of index property of the contravariant = metric=20 tensor.

With the exception of the locations of physical singularities, the = space-time=20 for the universe in which we live is an everywhere locally Lorentzian = spacetime.=20 A locally Lorentzian spacetime is a spacetime for which we can locally = transform=20 g_mn to h_mn where h_mn=20 is given by Eqn 2.2.4

A locally Euclidean Space-time is a spacetime for which we can = locally=20 transform g_mn to w_mn where w_{mn =}is=20 given by

(4.3.6)

In other words all the dimensions of a Euclidean "spacetime" are=20 spacelike.

Either type of spacetime can have Riemannian Curvature as these are = only=20 locally Euclidean, or Lorentzian.

46 Chapter 4 Starting GR

Note- Sometimes it is said that our Universe is everywhere locally = Euclidean.=20 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=20 say that our universe is everywhere locally Lorentzian.

Our universe is also described as being a globally Riemannian = spacetime. This=20 means that it globally takes the quadradic form of Eqn. 4.3.1

ds² =3D g_mndx^mdxⁿ

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=20 depend on speed, location with respect to gravitational sources etc... = nor upon=20 whose frame it was calculated from. Invariants are said to be invariant = to frame=20 transformations, or frame invariant. This does not imply that the value = of an=20 invariant must be the same everywhere (for example invariant = "densities") nor=20 that it must be conserved. In this context an invariant can be thought = of as=20 short for invariant scalar though there are tensor expressions such as = the delta=20 kronecker tensor whose elements are all frame invariant. Some people = also think=20 of tensors in general as invariants as they represent physical entities = and=20 physical entities will not depend in any intrinsic way on our choice of = frame.=20 From this perspective the "elements of a tensor" are thought of as = "projections=20 of the tensor" onto a coordinate dependent template. The paradigm for = this text=20 will instead be that the tensor is the template onto which the = projections have=20 been made. It is not invariant, but transforms according to the = transformation=20 properties of an infinitesimal displacement vector. Some relativity = authors use=20 the word scalar to be short for invariant scalars or what are just = called=20 invariants in this text. This is popular, but extremely inappropriate. = The=20 reason that it is inappropriate is that if people continue to redefine = things=20 without good reason so that they have a different meaning for whatever = theory=20 comes along then when they are used in general, eventually a student = will=20 practically have to learn a different dialect of the spoken language for = every=20 theory encountered. This is complication beyond reason. Here are a few = examples=20 of invariants

c The local vacuum speed of light
m Mass
p The pressure scalar [p =3D (1/3)(T^mnU_mU_nc^-2 - g_mnT^mn), for=20 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=20 consider the pressure of the gas after a balloon is popped in space. As = it=20 expands the pressure decreases and so it is not conserved.

An example from special relativity of a quantity that is conserved, = but=20 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=20 charge q.

4.3 The Metric and Invariants of GR 47

Consider the transformation of the full contraction of a tensor = T^m.

g'_mnT⁼²⁰'^mT⁼²⁰'ⁿ =3D [(=B6x^l/=B6x'^m)(=B6x^r/=B6x'ⁿ)g_lr][(=B6x'^m/=B6x^a)T^a][(=B6x'ⁿ/=B6x^b)T^b]

g'_mnT⁼²⁰'^mT⁼²⁰'ⁿ =3D (=B6x^l/=B6x'^m)(=B6x'^m/=B6x^a)(=B6x^r/=B6x'ⁿ)(=B6x'ⁿ/=B6x^b)g_lrT^aT^b

g'_mnT⁼²⁰'^mT⁼²⁰'ⁿ =3D d^l_ad^r_bg<= SUB>lrT^aT^b

g'_mnT⁼²⁰'^mT⁼²⁰'ⁿ=3D g_lrT^lT^r

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

Exercises

Problem 4.3.1

Find h_lrP^lP^r. If = the=20 contraction of a tensor is an invariant and this was a local result for = the=20 contraction of P^m, what does this = tell you=20 about mass in general relativity?

Problem 4.3.2

Write out P_m for g_mn =3D h_mn .

Problem 4.3.3

Recall problem 2.2.3. What does dt/dt' turn out to be for the=20 spacetime

ds² =3D (1 + az/c²)²dct² - = dx²=20 - dy² - dz²

Problem 4.3.4

Use Eqn 4.3.3 to find g^mn for = the=20 spacetime in Problem 4.3.3. Hint - this is simply a matrix = inversion.

Problem 4.3.5

Write out g^mn for

ds² =3D (A² + = g_zzb²(1 - f)²)dct² +=20 2g_zzb(1 - f)dctdz +=20 g_zzdz² - H²(dx² +=20 dy²)

Hint - g_mn is symmetric and = g^mn is a matrix inversion of g_mn.

Problem 4.3.6

Consider the spacetime

ds² =3D (1 - = r₀/r)dct² -=20 dr²/(1 - r₀/r) - r²(dq² + sin²qdf²)

Write out T_m for an arbitrary = vector=20 T^m and find T_mT^m.

______________________________________________________________________= _________________

48 Chapter 4 Starting GR

4.4 The Affine Connections and The Covariant=20 Derivative

We want to make equations for the general laws of physics out of = tensor=20 equations. So in developing a differentiation operator for general = relativity we=20 must assure that when it is operated on a tensor it results in something = that is=20 still a tensor. We find that many of the special relativistic laws of = physics=20 are described by equations involving ordinary differentiation and so = this=20 operator must also reduce to the ordinary differentiation operator in = local free=20 fall frames. Consider the chain rule for the ordinary differentiation of = a=20 tensor.

dT^l =3D (=B6T^l/=B6x^r)dx^r

Using the transformation property of a contravariant tensor we = find

dT^l =3D {(=B6/=B6x^r)[(=B6x^l/=B6x⁼'^=20
s)T'^s]}dx^r =

Using the product rule we come to

dT^l =3D (=B6²x^l/=B6x^r=B6x⁼²⁰'^s)T'^sdx^r + = (=B6x^l/=B6x'^s)(=B6T⁼'^=20
s/=B6x^r)dx^r =

And again from the chain rule we finally have

dT^l =3D (=B6²x^l/=B6x^r=B6x⁼²⁰'^s)T⁼'^=20
sdx^r + (=B6x^l/=B6x'^s)dT'^s

Now if on the right hand side we only had the second term then the=20 differentiation of a tensor would still transform as a tensor, but we = have the=20 extra first term so we know it does not. Thus to find a differentiation = operator=20 which maps tensors to tensors we introduce a second term in the = operation. The=20 new differential operator is called the covariant derivative=20 opperator.

DT^l =3D = dT^l + dT^l

(4.4.1)

For a contravariant vector the second term necessary to keep = DT^l a tensor is

dT^l =3D = G^l_mnT^mdxⁿ =

(4.4.2)

where the affine connection(sometimes called the Christophel = symbol of=20 the second kind) G^l_mn = is given=20 by

G^l_mn =3D (1/2)g^lr(g_mr,_n +=20 g_nr,_m - g_mn,_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

DT_l =3D = dT_l + dT_l

(4.4.4)

But here we have

dT_l =3D - = G^m_lnT_mdxⁿ =

(4.4.5)

In the case of the differentiation of a multiple mixed rank tensor we = find=20

DT^l..._k_... =3D dT^l..._k⁼²⁰_... + G^l_mnT^m..._k⁼²⁰_...dxⁿ +... - = G^r_knT^l..._r⁼²⁰_...dxⁿ -...

(4.4.6)

Also it is important to make note that though the affine connection = is a part=20 of a covariant derivative operator, it is not a tensor itself.

So, for example, the covariant derivative of a tensor T^l with respect to some invariant parameter = such as=20 dt is

DT^l/dt =3D dT^l/dt + G^l_mnT^m(dxⁿ/dt)

(4.4.7)

As mentioned, a comma will represent a partial derivative and a = semicolon=20 will represent a partial covariant derivative. So for example

T^l;_r =3D T^l,_r + G^l_mnT^m(=B6xⁿ/=B6x^r)

This simplifies to (4.4.8)

T^l;_r =3D T^l,_r + G^l_mrT^m

Exercises

Problem 4.4.1

Work out the affine connections and verify Eqn 4.3.2 for the = spacetime in=20 problem 4.3.3.

Problem 4.4.2

Given that the metric tensor is symmetric, verify that the Affine = connections=20 are symmetric in the lower indices. That is to say verify G^l_mr =3D G^l_rm.

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