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General Relativity : The conceptual premises, tensors, the metric and invariants, and Christoffel symbols.

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4.1 The Conceptual Premises For General Relativity [Einstein A., 1915]

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 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 according to an inertial frame, Lois's, 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.

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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 according to remote unaccelerated observers.

 

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 [Rindler, W., 1969]. 

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 physics the gravitational force f from a point active gravitational mass M1 acting on a point passive gravitational mass M2 [Kreuzer, L. B., 1968,] 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 [Roll, P. G., R. Krotkov, and R. H. Dicke, 1964] the gravitational mass M2 must be equivalent to inertial mass M2i [Braginsky, V. B., and V. I. Panov, 1971]. Then we have

ar = -GM1/r2

 (4.1.4)

In the following chapters we will have an invariant definition of mass similar to how we had defined mass as invariant in the previous chapters. There will also be a four-vector force equation in the form

where m is the mass as invariant.

Gravitation acting alone corresponds to . This yields:

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

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 General Relativity 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 mentioned 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?

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4.2 Tensors in General Relativity

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 has 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 rank two.

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...

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 General Relativity 41

Here we will define a tensor [Corson, E. M., 1953] in terms of its transformation properties. A contravariant tensor will be any quantity that transforms between frames according to

(4.2.1)

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

(4.2.2)

There are also mixed tensors. For example

(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 be a tensor describing something observed like say the electromagnetic field and is the four-vector velocity of the observer (c,0,0,0). It turns out that the electric field given by

is NOT a tensor. As is the four-vector velocity of whoever is the observer everyone uses (c,0,0,0) for himself as a result and the expression does not transform as a four-vector. . If were the four-vector velocity of one "particular" observer then the expression would transform correctly, but then it wouldn't represent the electric field to anyone except that observer and it would then only when is the electromagnetic field already expressed according to his own frame. Likewise the magnetic field

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

We must write the fundamental equations of physics as tensor equations such as

(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

Exercises

Problem 4.2.1

Use the 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 be a tensor describing something observed like say the electromagnetic field and is the four-vector velocity of the observer (c,0,0,0). It turns out that the electric field given by

is NOT a tensor. As is the four-vector velocity of whoever is the observer everyone uses for himself (c,0,0,0) as a result and the expression does not transform as a four-vector. . If 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 is the electromagnetic field already expressed according to his own frame. Likewise the magnetic field

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

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4.3 The Metric and Invariants of General Relativity

Recall that for the special theory, 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

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

The invariant interval will then take the form

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 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

(4.3.1)

For the general theory, when there is matter or fields of any type in the space it effects the form that 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 to . However one can always do the transformation locally.

We again express the invariant interval in the form

Given that the interval is invariant we know that

We also know that transforms according to the calculus chain rule

 

4.3 The Metric and Invariants of General Relativity 43

This results in

Now this is how a rank 2 covariant tensor transforms. Therefor if ds2 is to be invariant then 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 Newtonian physics the gravitational force or other fields were 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 the special theory we have

We can always transform to a local free fall frame according to which the metric is and its first order derivatives are zero, so we know so far that for a local free fall frame also

For such a local free fall frame, the Cristoffel symbols are zero so you have

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

(4.3.2a,b)

(Summation still implied on all four above)

Next consider the quantity

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

Rearrange terms

Yielding

Simplify

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

Simplify

 

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This yields

(4.3.3)

Contract this and we have

 

Which results in

(4.3.4)

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

and (4.3.5)

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

based on how they transform this becomes

Rearranging:

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

This simplifies to

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

 

4.3 The Metric and Invariants of General Relativity 45

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

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

But we've already verified that so we have

Which results in

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 to where is given by Eqn 2.2.4

A locally Euclidean Space-time is a spacetime for which we can locally transform to where 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.

 

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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

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 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

  1. c The local vacuum speed of light
  2. m Mass
  3. p The pressure scalar
  4. The proper time between events along a world line.
  5. 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 the special theory 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 General Relativity 47

Consider the transformation of the full contraction of a tensor

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

Exercises

Problem 4.3.1

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

Problem 4.3.2

Write out for .

Problem 4.3.3

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

Problem 4.3.4

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

Problem 4.3.5

Write out for

Hint - is symmetric and is a matrix inversion of .

Problem 4.3.6

Consider the spacetime

 

Write out for an arbitrary vector and find .

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4.4 Christoffel Symbols

We want to make equations for the general laws of physics out of tensor equations. So in developing a differentiation operator for the general theory 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.

Using the transformation property of a contravariant tensor we find

Using the product rule we come to

And again from the chain rule we finally have

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 tensor elements to tensor elements we introduce a second term in the operation. The new differential operator is called the covariant derivative opperator.

(4.4.1)

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

(4.4.2)

where the affine connection(sometimes called the Christoffel symbol of the second kind or in the context of general relativity often just Christoffel symbol for short) [Cartan, E., 1923] is given by

(4.4.3)

 

 

4.4 Christoffel Symbols 49

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

(4.4.4)

But here we have

(4.4.5)

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

(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 with respect to some invariant parameter such as is

(4.4.7)

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

This simplifies to (4.4.8)

(4.4.8)

And in terms of the lowered index we have

(4.4.9)

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

References Citations

Braginsky, V. B., and V. I. Panov, 1971, "Verification of the Equivalence of Inertial and Gravitational Mass, " Sov. Phys.-JETP 34, 464-466

Cartan, E., 1923, "Sur Les Varietes a Connexion Affine et la Theorie de la Relativite Generalisee," Ann. Ecole Norm. Sup. 41, 1-25

Corson, E. M., 1953, Introduction to Tensors, Spinors, and Relativistic Wave-Equations, Hafner New York

Einstein A., 1915a, "Zur Allgemeinen Relativitatstheorie," Press. Akad. Wiss. Berlin, Sitzber., 778-786, 799-801

Einstein A., 1915b, "Erklarung der Perihelbewegung des Merkur aus der Allgemeinen Relativitatstheorie," Press. Akad. Wiss. Berlin, Sitzber., 47., 831-839

Einstein A., 1915c, "Die Feldgleichungen der Gravitation," Press. Akad. Wiss. Berlin, Sitzber., 844-847

Kreuzer, L. B., 1968, "Experimental Measurement of the Equivalence of Active and Passive Gravitational Mass," Phys. Rev. 169, 1007-1012

Rindler, W., 1969, Essential Relativity: Special, General, and Cosmological, Van Nostrand, New York

Roll, P. G., R. Krotkov, and R. H. Dicke, 1964, "The equivalence of Inertial and Passive Gravitational Mass," Ann. Phys. 26, 442-517

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