12.1 Metric Engineering
If we limit our physics to Einstein's special relativity theory, accomplishing interstellar flight is virtually a technological impossibility. Distances between the stars are too large. As far away as most of the interesting stars are, even if one could accelerate a ship with a high enough acceleration to make a round trip within a crews life time the huge accelerations would crush the crew against the hull of the ship. Even if this crushing problem were overcome, the speed c becomes a cosmic speed limit so the round trip may wind up returning the crew generations after they left according to the time of the earth. Also, the fuel requirements to produce such accelerations are impossibly high. All these problems can potentially be done away through a new field of research called metric engineering. Spacetime's differential geometry given by the metric tensor is responsible for how all things in the universe tend to age and move. The differential behavior that the metric can take is also determined by what matter is in the space according to Einstein's field equations. It is possible to artificially engineer different metrics by choosing particular arrangements of matter. It is possible that we will be able to engineer a metric that will change the remote coordinate speed of light around starships thus removing the speed c limitation, or to engineer a metric that manipulates inertia itself making a state of coordinate acceleration the natural state of the ship instead of a state of constant velocity. This would eliminate fuel requirements altogether as well as eliminate the problems of the crew becoming crushed from the acceleration of the ship.
Here's a simple example of metric engineering for theoretical general relativistic physics.
Space is not truely vaccum, but is actually filled with fields and along with their existence also virtual particles. This means that the so called vacuum of space actually has a nonzero energy density even though it is usually taken to be the "zero point" for energy density. Now consider if we might polarize the vacuum so that some of the energy from one spot is moved to another spot. Now we have one spot that has a positive energy density relative to the zero point and another spot with a negative energy density relative to the zero point. This second spot is called a hole. Now the positive
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energy density region will have the properties of a mass. It will attract all things toward itself, including the hole. On the other hand, the hole would have the properties of a negative mass. It would repel all things from it including positive mass. As a result the positive mass accelerates in the direction away from the hole and the hole is drawn by the positive mass to chase it. The natural state of this system is a state of coordinate acceleration instead of a state of constant velocity with respect to remote inertial observer frames. The cabin of a ship could be placed between the positive mass and the hole and it would be drawn to accelerate along with the system. The crew would be drawn to accelerate away at the same rate as the cabin and so they would not be crushed against the walls of the ship.
Consider a positive mass M at r1 and a negative mass -M at r2. Find the acceleration of the system and show that total momentum and energy are conserved.
There are more than one type of wormhole in theory for general relativistic physics. First we'll look at the type associated with Schwarzschild black holes. We'll second look at those associated with Kerr-Newman black holes. Third we will look at the Morris-Thorne wormhole. Finally we will consider a modification of the Morris-Thorne wormhole geometry abandoning spherical symmetry to arbitrarily reduce the exotic matter requirement.
The Schwarzschild spacetime's interval expressed in a remote observer's spherical coordinate system takes the form of Eqn.10.1.1
This has a coordinate singularity where the dr2/(1 - 2GM/rc2) term becomes infinite at
r = 2GM/c2. This infinity problem can be removed by a transformation to a different choice of spacetime coordinates. There are various transformations that do this, but a particular choice of transformation called the Kruskal-Szekeres transformation revealed something previously unknown about the black hole spacetime geometry . For our external region of space the Kruskal-Szekeres coordinate transformation is
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R is a constant.
The transformation above is only valid down to the event horizon. Beyond this an extension must be done in which the transformation becomes
With this transformation the equation for the invariant interval becomes
Note that if we choose R = 4GM/e1/2c2 then for r' ~ r ~ 2GM/c2 the interval becomes that of Einstein's special relativity theory with the exception that it's in spherical coordinates. Thus this could serve as a transformation to a frame local to the event horizon. However, we must transform to a frame where the affine connections also vanish there to have a local free fall frame.
Now if we consider working out the inverse transformations to these we must note that the squared r' and ct' each have two roots. This leads us to realize that the Schwarzschild spacetime of general relativity actually has two, not one external regions and two, not one, internal regions, each corresponding to our choice of sign. The inverse transformation for our external region is
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The internal region referred to as the black hole internal region has the inverse transformation given by
The second internal region corresponding to the other choice of sign has the inverse transformation given by
This region has become termed the white hole interior region.
The second external region has the inverse transformation given by
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The spacelike hypersurfaces extending through from one external region to another are called wormholes.
At this point the question arises, whether it is possible for objects or information to transverse the wormholes from one external region to the other. There are two potential problems with this being argued to date. First off, is the question of wormhole stability. Second, and really the only relevant problem is that in this spacetime there are no time-like or light-like geodesics connecting the two external regions. The argument over wormhole stability goes as follows. A hypersurface that is apparently space-like in the Schwarzschild (r,ct) coordinates is never space-like for the internal regions in the Kruskal-Szekeres (r',ct') coordinates. As a result any wormhole "static" in the (r,ct) system is always dynamic for the interior regions in the (r',ct') system. This leads to a short life-span of the connection between any two external regions according to the (r',ct') frame. The life-time of the wormholes are argued to be to short lived for information to be communicated between the two external regions.
The real problem with transversing a wormhole of general relativistic physics is that none of the geodesics connecting these two external regions are time-like or light like. For information to cross from one external region to the other, it would have to follow a space-like path during at least part of the journey. In other words, in order for information to cross from one side of a wormhole to the other without winding up hitting the physical singularities it would have to travel faster than light in a way not allowed even by general relativistic physics. If information could do this, then the first problem of stability wouldn't be a problem at all as the information could travel through the wormhole arbitrarily fast and make it through before the wormhole connection was broken. This is why the second problem is the real problem with transversing this type of wormhole.
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Another way to conceptualize a wormhole topology goes as follows.
The Schwarzschild spacetime's interval expressed in a remote observer's spherical coordinate system takes the form
Consider a hypersurface given by ct = constant and for simplicity a slice running through the equatorial plane at resulting in
One does so by imagining an extra-dimensional flat space in which this surface is imbedded.
This results in
The anti-derivative of which gives
w2 = 4r0(r - r0)
Sweeping this around the w axis to include all results in a paraboloid surface
We usually pick the upper half of the surface to represent our external region. The lower half is used to represent the other external region previously discussed.
Such a wormhole surface is also sometimes called an Einstein-Rosen bridge.
Next we will consider the Kerr-Newman black holes(charged-rotating black holes) that have geodesic paths between certain external regions that are likely to be transversable, at least going one way.
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The next is a conformal diagram for the maximal extension for a Kerr-Newman space-time(a charged rotating Black Hole).
The curve W is a spacelike hypersurface connecting two different external regions and the question of the stability of a connection allowing this path to form is as valid for this spacetime as it was for the Schwartzschild wormholes. Even if the different regions do reach a stable connected state we still have a problem trying to transverse a path between region I and region IV. That problem is that no timelike paths connect the regions. However, There is a time-like geodesic from region I to region VIII or IX. These paths are likely to be transversable. Since these are timelike geodesic paths and not a spacelike hypersurface, they are not called wormholes, but I see no problem with qualifying the word and calling such a geodesic path a timelike wormhole. This is a different aspect of the black hole geometry and so the problem of wormhole stability really does not apply. One might argue that region VIII or IX is nothing more than a mathematical repeat of region I. If this is the case, then the path is a closed timelike loop and a probe dropped into the hole would fall back out to the same point in space and time from which it started. This presents a grandfather time paradox. With a little boost along, or at the beginning of the path, the probe comes out to arrive at the same point in space from where it was thrown in even before the time it was released! However if region VIII and IX do turn out to be other universes or if the event horizon between region VII and VIII or IX corresponds the outer horizon at the birth of what appears to be another hole far removed from the one entered, then such a causality paradox can be avoided. If it is the case that this is indeed an exit into another universe or location, then we should also note that the trip is likely to be one way only. One can only travel "up" along the diagram. This is probably where the TV series Stargate SGI came up
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with the concept of outgoing wormholes Vs incoming wormholes. Each stargate can only dial out and once you've dialed out, as long as that wormhole stays open, no one can come through from their end to yours.
We next turn to the Morris-Thorne wormhole. The Morris-Thorne Wormhole can be expressed by the following interval.
is just a gravitational potential allowed by the spacetime geometry. One falling along the -w direction falls "inward" toward the hole with decreasing r until reaching the minimal value of r. This minimal value is the radius of the wormhole throat. Continuing along the -w direction, one finds himself falling "outward" again away from the hole again with increasing r. This type of wormhole has great advantages over the spacelike wormholes associated with black holes previously discussed. For instance, there is no event horizon associated with this wormhole. The coordinate speed of light is nowhere zero. Because of this, remote observers can observe travelers cross from one external region to the other. Also, with this kind of wormhole timelike geodesics connect the different external regions. So one needn't travel at locally faster than c speeds to transverse the wormhole.
The stress energy tensor required for this exact spacetime is
The major problem with arbitrarily cooking up this wormhole geometry is that the stress-energy tensor's energy density term T00 has a negative energy density matter part that dominates the other parts at the wormhole's throat corresponding to the -2rd2r/dw2 in the numerator. As such you will come across many places talking about stable wormholes concluding that negative energy matter or "exotic matter" is therefor required to form a stable wormhole.
As for a simpler example calculation consider the case:
which yields a negative energy density term of
The F(w) can be chosen to arbitrarily reduce the magnitude of this energy density, but at the cost of introducing a gravitational acceleration field around the wormhole, so lets modify the wormhole, in particular lets not demand spherical symmetry, and see how this effects the stress-energy tensor. Such a modification may take the form:
For this example lets look at the case
It turns out that significantly impacts all of the nonzero stress energy terms and in fact as it goes to this line element goes to a flat spacetime vacuum solution:
This limit as is vacuum and so completely eliminates all exotic matter. Thus by not requiring spherical symmetry for the wormhole you can arbitrarily reduce the exotic matter required by this kind of wormhole. In short this is a stable wormhole geometry that one would cross directionally more like the wormholes of "stargate".
It turns out that 12.2.15 is a nonzero angular momentum parameter, but zero mass limit of what the Kerr solution becomes. Since it is a zero mass limit of the Kerr-solution, apparently the Kerr solution has a stable transversable wormhole geometry within it, but has a curvature of mass term embedded into it. So in contrast to the usual proposal of constructing a large wormhole by taking a small quantum mechanically produced wormhole and feeding it exotic matter until it grows to the size you want with spherical symmetry, I propose that one instead take an over extreme rotational micro wormhole and look for a way to increase its rotation, but not its mass until "a" is large, but its mass is not so large as to make escaping the gravitation a difficulty.
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How would the Morris-Thorne wormhole appear from an outside observer?
How would the wormhole of equation 12.2.14 appear from an outside observer?
What is the coordinate speed of light described by Eqn.12.2.3? Draw a path from one external region to another on figure 12.2.3 that does not violate this speed limit and does not intersect a physical singularity
What is the coordinate speed of light moving along w at constant and described by Eqn.12.2.14? What does this tell you about a remote observer's ability or inability to see something transverse the wormhole in either direction?
12.3 Time travel
In this section we investigate the possibility of time travel (the sci-fi sense of the term) within the realm of Einstein's theory of general relativity. We already know that in the physics of Einstein's special relativity theory faster than c information transfer would lead to this kind of time travel. It is because of this that it is often said that Einstein's theory of special relativity implies information can not be transferred faster than c. Since this page is about the physics of Einstein's general relativity theory we are not concerned with this method of time travel. First we will look at a possible form of time travel associated with black holes. Second we will look at the possibility of a the Morris-Thorne-Yurtsever wormhole time machine. Finally, we will look at the possibility where metric engineering is used to construct a H.G. Wells like time machine.
Refer to figure 12.2.3. As has been shown in section 11.1, an object dropped radialy along the polar axis into the hole gets thrown back out in a finite proper time. On the above graph this motion is described by a path that that is a vertical line going from region I through region II then VI then VII and back out into region IX. Region I is the external region representing our universe outside of the hole. As discussed in section 12.2, Region IX may be another universe all together or the path may actually come out an opening into our same universe from a far removed hole. However, Region IX may also just be a mathematical repeat representative of the same universe from which the object entered. If this is the case, then since the probe falls back out to the same remote observer coordinate from which it is dropped we find that a closed time-like loop is formed. The probe returns to the same point in space and time from which it is dropped according to the remote observer coordinates. With a boost along the path it can even return to the point in space from which it was dropped at a time before it started its journey.
Using the definitions of and eqn 11.1.42 and eqn 11.1.43 for motion in a Kerr-Newman spacetime (the spacetime for a charged rotating black hole) of general relativity, it is easy to work out the time travel equation. This is
In this idealized space-time one can actually travel as far back in time as one desires. In a realistic space-time in which the hole has a beginning where it is formed under the gravitation of a rotating star, one can only travel as far back in time as the hole existed. Thus, if metric engineering is used to construct this type of time machine, a time traveler can not travel to a time prior to the time machines creation. However, one might travel far out into the galaxy and find a
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very old black hole to use for travel as far back as it existed.
This type of time travel does lead to grandfather paradoxes and as a result physicists have felt compelled to create new objections to the possibility. As we've discussed region IX may be representative of another universe altogether in which case there is no grandfather paradox, but other objections have been offered. Some have suggested that in such cases where general relativity seems to allow this kind of time travel, the universe might be ordered in such a way that there will actually be some quantum mechanics effects that arise to negate the possibility. One type of grandfather paradox that superficially seems to be implied by the realization of the possibility is that there is a point in between the outer and inner event horizon where the in falling object seems to collide with its future self. Fortunately, when we study the graph we see that this needn't occur because this "single point" actually exists in two separate locations in the maximal extension. One is in region II and the other is in region VII.
One possible way out of the logic grandfather paradox that still allows for this sci-fi type of time travel is as follows. Region I and region IX do look graphically like separate universes. If they are not altogether separate universes or separate external regions, they may still be separate phases of the same universe. In quantum mechanics, the state of a system is described by a wave function. Where continuum states are concerned, the wave function is comprised of an infinite number of phase waves. These are often called Eigenstates, but more often so when considering discrete states. Though quantum mechanics works as far as making experimental and observational predictions is concerned, there are various different ways of interpreting quantum mechanics. One of these is the many worlds interpretations. In this interpretation each phase state corresponds to an actual reality and when a determination is made as to which state is actually measured, these different phases of reality split apart leaving us with many worlds. If region IX is actually a different phase of the same universe, then we can get around the logic grandfather paradox in the following way. At some point in a remote observer's time there is the possibility of an arrival of a time traveler from the future who intends to kill his grandfather before his father is conceived. This point is where the phases split. In one phase, no time traveler arrives and so in this phase the time traveler is born. This time traveler leaves this phase of the universe for the original point in time, only in the other phase of the universe. In that phase of the universe, the time traveler kills his grandfather before his father is conceived and so in that phase of the universe there is no time traveler born to kill his grandfather in this phase of the universe. In this way the grandfather paradox is not disastrous or a threat in any way. Instead, the different phases are not only not a danger to each other, but they each seem to be responsible for the form the other takes.
Next consider a Morris-Thorne-Yurtserver type wormhole time machine of general relativity. Due to a quantum mechanics uncertainty principle, there are large energy fluctuations in the vacuum at very small scales. These fluctuations are large enough to produce micro-wormholes. If one could locate such a wormhole and feed it what energy it needs to enlarge it to our macro world's scales then one may use it as a quick portal shortcut from one end to the other. It could be used to travel or communicate effectively "instantaneously" from one end to the other. Note-the simultaneity is according to an observers frame for which the observer is still with respect to one entrance of the wormhole. The wormhole time machine assumes the possibility that both ends open into the same universe. Consider the possibility that one end of the wormhole remains on earth while the other entrance moves away carried inside a ship moving off at a constant rate. If an observer on earth watches the clock on the ship by looking through a telescope at it instead of through the wormhole of general relativity, ordinary time dilation will be observed to take place. Likewise, if the ship observer watches the earth's clocks through a telescope instead of through the hole ordinary dime dilation is observed in the traditional twin paradox fashion. For simplicity, lets say the clocks were synchronized at the instant the ship and earth were together. Now if a signal is sent through the wormhole from the earth to the ship then it arrives virtually instantaneously according to the earth's externally inertial frame. Due to the time dilation the signal will arrive according to the ship's clock some time readout earlier than the earth's clocks read. Lets say that the ship immediately returns the signal through the wormhole. A symmetry is then assumed in which the return signal will also be received back at earth virtually instantaneously according to the "ship frame". This assumption may be unphysical and may be what invalidates the scenario which we will come back to. If the assumption is not unphysical, then because the ship observed the earth clocks to run slow due to time dilation the return signal will be received at an Earth clock time readout even before this. In that case, the signal will arrive back at earth at a time before it was sent from earth. Now we return to the symmetry assumption that was made. Consider the possibility that such a wormhole only formed so that each end is initially stationary with respect to the other. To set up the time travel paradox we have to accelerate one end of the wormhole. This sets up an intrinsic asymmetry between the history of each wormhole entrance. As a result if communication is instantaneous from entrance A to entrance B according to an observer at rest at entrance A, then the asymmetry in the history of the two entrances may produce an asymmetry in the result for a signal traveling from B to A. In that case communication from B to A through the wormhole may not be instantaneous according to an observer at rest with respect to B at all. Simultaneity may become direction dependent. For example, looking through the wormhole from A at B, the earth frame may observe the same ship clock readout as observing it without looking through the wormhole. Whereas looking through the wormhole from B at A, the ship observer may observe a time on earth's clock advanced of the time observed looking along another path. Again, the asymmetry would be due to the acceleration of one entrance of the wormhole from an initially static state. This situation would prevent a grandfather paradox.
The final type of time machine time travel we will discuss makes use of metric engineering to manufacture a desired spacetime geometry. This is something along the lines of a H.G. Wells type of time machine. Consider the following metric
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This as it stands is a vacuum field solution with a zero Riemann tensor. T(ct) appears because it corresponds to a frame in which we chose to vary how our clocks run mechanically. Because of this, if we choose T(ct) = 1 to correspond to o forward running clock, then T(ct) = -1 corresponds to a clock running backward. So far there is no sci-fi like time travel associated with this metric. Next we will make a modification to the metric as follows.
If we have df/dr = d2f/dr2 = 0 at r = 0 and at , and f = 0 at r = 0, and f = 1 at , then this metric will be a vacuum field solution with a zero Riemann tensor at these locations. The interesting aspect of this metric is that if then an observer at the origin will observe that forward running clocks at will actually run backwards. Unfortunately there is one severe physical problem with this kind of time machine metric. The stress energy tensor diverges on the event horizon ( where we have (1 - A)f(r) = 1 ) where there is a singularity in the contravariant metric tensor.
While A is a constant, we can more simply express the above interval with the form
ds2 = [T(r)]2dct2 - dr2 - r2(dq2 + sin2qdf2)
The exact solution for the Einstein tensor is
And the stress-energy tensor's solution is given by Einstein's field equations
The terms diverge because T(r) is in the denominator and T(r) must go to zero at the event horizon.
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So let us seek to modify this so that the solution corresponds to a mere electric field exact solution across the horizon and place the matter responsible for the field under the horizon so that the divergence of terms will be known to be a mere artifact of using coordinates that are singular there. Consider the following metric
This is merely a frame transformation of an extremely charged Reissner-Nordstrom solution. Now consider instead of putting the charge at the origin, putting it at r = R. Then the solution becomes
Notice that this ds2 now has a g00 with the same boundary conditions as 12.3.3b. Unlike uncharged matter collapsing without hope of escape into a Schwarzschild black hole geometry, the exact solution for the extreme charge state of the matter allows the matter to remain without collapse at any location, including the location chosen here which is half of the event horizon radius according to this r coordinate. This yields a H.G. Wells style time machine solution where one can sit inside the time machine where everything is normal, while time runs backwards outside an event horizon, even though the amount of matter required to make one of the size of what he had in his book is on the order of a star.
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Compute the Einstein tensor and the Ricci-scalar for equation 12.3.5 in order to show that the corresponding stress-energy tensor is
and that the Ricci-scalar is zero aside from r = 0 where the mass-charge is for that line element. GRTensorII is a free downloadable tensor calculus package that is good for this kind of work.
Describe how Eqn. 11.1.23 relates to figure 12.2.3.
Draw lines of constant remote observer coordinate time extending to all regions on figure 12.2.1
12.4 Gravitational Field Propulsion
Consider the following stress energy tensor.
This has as its weak field approximate solution:
So let us consider the following scenario for and , where is some negative constant and .
Inside the region a test particle free floats, but for the solution becomes the Rindler spacetime:
The concept is depicted in the figure:
In this scenario an observer free floating inside experiences no force, but observes the remote stars to behave as if he had a proper acceleration of .
As such general relativity has the implication that if one could produce the stress-pressure terms of 12.4.1, one would be able to accelerate without feeling it amongst the remote stars to any speed less than c indefinitely. What's required to do this is determined by the stress-pressure terms, but notice that the energy density term does not determine the proper acceleration. An unusual stress-pressure state, but NOT "Negative energy" matter is what is required.
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12.5 Relativistic Unification Theories
Here we will take a look at Einstein's unification theory, then look at Kaluza-Klein unification which serves as the springboard into string theories, and finally look at how General Relativity itself can be worked as a classical theory for which gravitation is already unified with electromagnetism.
In doing pre-relativistic physics we often think of the electric and magnetic "fields" as two separate vector fields. We express the electric and magnetic field vectors as two separate three-component vectors.
It turns out that these pseudo-vectors taken separately do not transform like the vectors of relativity. By themselves, they do not transform like tensors. It turns out that these are actually incomplete components of the actual field vector known as the electromagnetic field tensor. In local Cartesian coordinates the electromagnetic field tensor is Eqn. 6.3.11
[note - the electric field is given by
and if one inserts the unprimed frame observers four-vector velocity then it can be written as a product of tensors
, but even so is still not a tensor. As is the four-vector velocity of whoever is the observer, everyone would use for his own 4-velocity , as a result and then 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 given by
which can then be written in terms of the unprimed frame observers own four-vector velocity as
is also not a tensor. An easy way to conceptually prove that it is not is to imagine a proton beam's magnetic field and note that it vanishes according to the proton frame. A tensor can not be transformed away.]
In finding this electromagnetic field tensor it was discovered that the electric and magnetic "fields" were actually incomplete components of a single unified electromagnetic field. As gravitation and other fields have similar characteristics it is thought that all of the fields that we think of as separate are actually all just different aspects of a single potential or unified field. It is commonly known that after the core development of Einstein's general relativity theory, Einstein continued to work on a unification theory for unification of the gravitational and electromagnetic "fields". What is not commonly known is that he did have success in developing the basis for such a unification theory. However, most modern attempts at finding a good unified field theory are rooted in Kaluza-Klein unification or in quantization unification because people have been able to produce results more descriptive of a wider range of particle fields characteristic of the observed nature of the universe.
In relativity, the metric tensor plays the role of a gravitational potential. Outside of Einstein's unification the metric tensor is completely symmetric. We note that the electromagnetic field tensor is completely anti-symmetric. In Einstein's unification theory, the metric tensor can be written as a sum of a symmetric and an anti-symmetric part, but Eqn.4.4.3 will remain valid. This also leads to an affine connection that can be written as a sum of a symmetric and an anti-symmetric part.
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Einstein then defined a tensor density
detg = the determinant of the metric tensor.
This tensor density then also can be written as the sum of a symmetric and an anti-symmetric part.
The Lagranginan density for his theory is and his field equations are
(the free-space gravitational field eq)
The latter implies
This generates Maxwell's curl equations when we make the relations
E0 is a parameter with units of electric field.
Making this final relation we see that the gravitational and electric and magnetic "fields" all come from the same unified source(the metric tensor). The metric tensor gives rise to the affine connections that we interpret as a gravitational field and it also gives rise to the components of the density tensor that we interpret as electric and magnetic fields. In fact is the dual tensor Eqn. 7.1.7 of the electromagnetic field tensor above.
Kaluza-Klein theories take a different approach to unification of the fields then Einstein's approach to unification. These introduce extra dimensions to the spacetime and the electromagnetic potential is related to the elements of the metric tensor corresponding to the extra dimensions. String theories were born from this train of thought. Here we will present the
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basis of the actual Kaluza-Klein theory. First we introduce the five dimensional metric tensor.
We distinguish it from the ordinary four dimensional metric tensor by putting a bar underneath.
Though the indices include a fifth dimension for , the metric and all equations are written only as functions of the four dimensions which we can normally observe. For the ordinary metric tensor is related to the five dimensional metric tensor by
Here, one element is taken to be a given,
The electromagnetic field tensor is then written in terms of an electromagnetic potential.
And the electromagnetic potential is related to the five dimensional metric tensor.
V0 is merely included here as a parameter of the theory with units of voltage.
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The Lagrangian density for this theory is.
This results in the Einstein field equations for electromagnetism "alone" and an otherwise vacuum region becoming
From this,. But this is related to the electromagnetic field tensor by . Which results in . This is comparable to the section on Maxwell's equations Eqn. 7.1.5 for .
Finally lets look at how General relativity serves as a classical already unified theory of gravity and electromagnetism.
In section 1 of chapter 7 we found that for the exact solution to Einstein's field equations for an arbitrary static matter distribution of extremal charge we could write the rank 2 electromagnetic field tensor in terms of spacetime geometry as
Where was the timelike killing vector. This infers that the electromagnetic field is a result of the spacetime geometry just the same as gravitation is a result of spacetime geometry. It allowed the equation of motion for a test charge to be expressed as a function of spacetime geometry alone as
For the killing vector acting as the vector potential is the spacetime geometry statement that there are no magnetic monopoles and guarantees that the covariant gradient of the killing vector obeys general relativity's version of Maxwell's magnetic field equations.
Since we now know the electromagnetic field for the spacetime in terms of spacetime geometry we can write an invariant lagrange density for the electromagnetic field in terms of spacetime geometry
(some notations call this the lagrange density squared, but for this chapter I'll define it this way)
The overall sign comes from a choice of convention. The spacetime was static and isotropic which means that it is likely that the killing vector T for this lagrangian having the direction (1,0,0,0) was a unique case for this solution and more generally will take other directions and have coordinate dependence.
In order to develop a quantized unified theory of gravity and electromagnetism from here we need to hypothesis a combined electromagnetic lagrange density in terms of spacetime geometry. So here I write down an invariant lagrange density as a postulate for an electromagnetic field that depends only on the spacetime geometry and on the current's velocity four-vector.
Where are rank 1 killing vectors. And a b f and g are invariants. The inclusion of a Ricci-scalar term allows a test charge to have a reaction to entering regions of space where nonzero mass is located. To get the lagrangian you then integrate with an invariant volume element over all space and add where mg is the passive gravitational mass.
Then from the results we then get the lagrangian mechanics version of Newton's 2nd law
We can then find directions of isometry so that transforming to coordinates that lay in the directions of isometry the term vanishes for changes along those direction in what cases we have conserved parameters of
At which point we can make the replacements
As a second postulate for a quantum unified field theory. And then the eigenstate wave equation becomes
To yield a quantized unified theory of gravity and electromagnetism.
Considering Kaluza-Klein theory, explain how in accordance with parallel transport, Eqn. 12.5.8 and expression of the metric components in terms of the ordinary dimensional coordinates only demands rotational invariance about the extra-dimensional axis.
In Kaluza-Klein theory the metric tensor is the potential for other fields in nature besides the gravitational field. How is the relation between the electromagnetic field and the metric tensor different in Einstein's unification theory?
Find the expressions for E0 and V0 in terms of known physical constants.
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Boyer, R. H., and R. W. Lindquist, 1967, "Maximal Analytic Extension of the Kerr Metric," J. Math. Phys. 8, 265-281
Broeck, C. V. D., 1999, "A 'Warp Drive' With More Reasonable Total Energy Requirements," Class. Quant. Grav. 16, 3973-3979
Callaway, J., 1953, "The Equations of Motion in Einstein's New Unified Field Theory," Phys. Rev. 92, 1567-1570
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