Jack Sarfatti

Modified Theories of Gravity: Traversable Wormholes http://bit.ly/ojY86N

Quotes from Miguel A. Oliveira

(Submitted on 13 Jul 2011)

"This MSc thesis is divided in to two parts. The first, covers the foundations of theories of gravitation, and, the second incorporates original work on the subject of the existence of traversable wormholes in $f(R)$ modified theories of gravity.

A short incursion in the field of scalar-tensor theories had to be made, owing to an apparent inconsistency in the result previously found."

"In this work, we consider these modifications, but focus on the existence of a specific type of exact solution: traversable wormholes. These are hypothetical tunnels in space-time, and are primarily useful as “gedanken-experiments” and as a theoreticians probe of the foundations of general relativity, although their existence as a solution of the field equations may be regarded as a viability condition of the theory."

"However, despite this astonishing success, the mere fact that GR is a scientific theory makes it provisional, tentative or probational! Inevitably, provided scientific research follows it’s normal course, some piece of informa- tion (experimental or otherwise) will come along, that doesn’t easily fit within the framework of any given theory, regardless of how well constructed it may be. For General Relativity, the acceleration of the universe, — prompting the introduction of dark energy —, the rotation curves of galaxies and the mass discrepancy of clusters of galaxies — supporting the existence of dark matter (see [1] for a review), and finally, on a more fundamental plane, GR’s obstinate resistance to all attempts at it’s quantization [2], constitute a set of instances that strongly motivate the introduction of Modified Theories of Gravitation (MTG’s)."

"Summarizing, Einstein’s approach was embodied in heuristic principles that guided his search from the beginning in 1907. The first and more lasting one was the ‘Equivalence Principle” which states that gravitation and inertia are essentially the same. This insight implies that the class of global inertial frames singled out in the special relativity can have no place in a relativistic theory of gravitation. In other words, Einstein was led to generalize the principle of relativity by requiring that the covariance group of his new theory of gravitation be larger than the Lorentz group. This will lead him through a long journey and in his first step, already in his review of 1907, Einstein formulated the assumption of complete physical equivalence between a uniformly accelerated reference frame and a constant homogeneous gravitational field. That is, the principle of equivalence extends the covariance of special relativity beyond Lorentz covariance but not as far as general covariance. Only later Einstein formulates a Generalized Principle of Relativity” which would be satisfied if the field equation of the new theory could be shown to possess general covariance. But Einstein’s story appealing to this mathematical property, general covariance, is full of ups and downs."

"Also, around the same time (1920), there were discussions about whether it is the metric or the connection, that should be considered the principal field related to gravity. In 1924, Eddington presented a purely affine version of GR in vacuum. Later Schro ?dinger, generalized Eddington’s theory to include a non-symmetric metric [10]. These are vacuum theories and, great difficulties are encountered when any attempt is made to include matter into them. It is worth mentioning here, one other approach to this question, that is, to have both a metric and a connection that are at least to some extent independent. A good example is the Einstein-Cartan theory, that uses a non symmetric connection and, Riemann-Cartan spaces. This theory allows the existence of torsion and relates it to the presence of spin."

"We could also have, in addition to the tensor field two others, namely, a scalar and a vector one, which is the case of Bekenstein’s Tensor-Vector-Scalar (TeVeS) theory proposed in 2004 [11]. It has a curious motivation: to account for the anomalous rotation curves of galaxies, Milgrom proposed to avoid introducing dark matter by changing Newton’s laws [12], this is called Modified Newtonian Dynamics (MOND); since this theory is not relativistic, TeVeS was crafted to be the relativistic extension of MOND. Einstein-Aether theory is another theory of this kind. In this one, a dynamical vector (but not a scalar) field is added. The aether is a preferred frame, (whose role is played precisely by this vector field), that would have to be determined on the basis of some yet unknown physics. It is interesting to note this frame may lead to Lorentz invariance violations."

"The search for quantum gravity, has produced a theory, known as string theory, (which we will simply describe as a perturbative, and hence background- dependent theory, that uses objects known as strings as the fundamental building blocks for interactions), this is believed to be a viable theory unifying all (four) physical interactions. One of the predictions of this theory is the existence of extra spatial dimensions. Moreover, recent developments in string theory, have motivated the introduction of the brane-world scenario, in which the 3-dimensional observed universe, is imbedded in a higher-dimensional space-time. Although, most brane-word scenarios — notably the Randall- Sundrum type — produce ultra-violet modifications to General Relativity, i.e., extra-dimensional gravity dominates at high energies, there are those that lead to infra-red modifications, i.e., those in which extra-dimensional gravity dominates at low energies."

"A different class of these brane-word scenarios, exhibits an interesting characteristic, that consists in the presence of the late-time cosmic acceleration, even when there is no dark-energy field. This exciting feature is called “self-acceleration”, and the class of models where it arises, is named the Dvali-Gabadadze-Porrati (DGP) models. It is important to note, however, that these DGP models offer a paradigm for nature, that is fundamentally different from dark energy models of cosmic acceleration, even those with same expansion history [1]."

"The Gauss-Bonnet invariant is also important because recent developments in String/M-Theory suggest that unusual gravity-matter couplings may become important at the present low curvature universe. ... Finally, since we mentioned string theory, we feel obligated to refer, if only briefly, it’s main alternative, that is, Loop Quantum Gravity (LQG). LQG, unlike string theory, is only (and this is already a lot), a quantum theory of gravity, it is not a unified theory of physics."

"It also does not predict the existence of extra spacial dimensions. LQG is an example of the canonical quantization approach to the construction of a quantum theory of gravity. It is a fully background-independent and non-perturbative quantum theory of gravity [2]. There are no experimental data whatsoever, supporting or disproving, any of these two theories. For the moment, all we have to compare them are consistency checks and aesthetic principles."

"It has been pointed out [3], that the so called, experimental tests of General Relativity (namely, the deflection of light; the shift in the perihelion of Mercury; and, the gravitational redshift of distant light sources eg. galaxies), are in effect, tests of the underlying principles, rather then tests of the theory itself. ...

Dicke proposed that the two following basic assumptions should be included[14]:

1. The set of all physical events is a 4-dimensional manifold, called Space- time;

2. The equations are independent of the coordinates used – Principle of Covariance.

This is called the Dicke Framework."

*"Coordinates" really mean local frames of reference, i.e. tiny detectors of light and particles. The transformations are always between locally coincident detectors separated from each other spatially and temporally by distances and times small compared to the inverse square root of the local curvature field tensor components. - wrote Jack Sarfatti*

"Wald [15] give a different definition of covariance and distinguish between:

• General Covariance: there are no preferred vector fields or no preferred basis of vector fields pertaining only to the structure of space which appear in any law of physics.

• Special Covariance: if O is a family of observers and, O′ is a second family obtained from the first by “acting” on it with an isometry, then if O makes a measurement on a physical field, then O′ must also be able to make that same measurement. The set of physical measurements are the same for the two families.

Dicke added two more requirements, that gravity should be associated with one or more fields of tensorial nature (scalar, vector or tensor), and that the field equations should be derivable from an invariant action via a stationary action principle."

In addition to the above notion of "general covariance" i.e. comparing locally coincident small detectors each on an arbitrary world line (including accelerating detectors), Einstein introduced a second organizing principle: "a version of the principle of equivalence was already present in Newtonian Mechanics. It was even mentioned in the first paragraph of Newton’s Principia (see, [14] page 13 and Figure 2.1 there). In that context, this principle states that the inertial and passive gravitational masses are equal, mI = mp,1. In this form the principle can also be equivalently stated in the following way: all bodies fall (when in free-fall) with the same acceleration, independently of their composition and mass."

"Several current forms (and several different formulations throughout the literature) of the equivalence principle may be distinguished [3]:

• Weak Equivalence Principle (WEP): the trajectory of an uncharged test particle (for all possible initial conditions ) is independent of its structure and composition;

• Einstein Equivalence Principle (EEP): this assumes that the WEP is valid and, that non-gravitational test experiments have outcomes that are independent of both velocity (Local Lorentz Invariance - LLI), and position in space-time (Local Position Invariance - LPI);

• Strong Equivalence Principle (SEP): this assumes that the WEP is valid both for test particles and self-gravitating bodies, and also assumes LLI and LPI for any local test experiment."

"The EEP allows us, (after rescaling coupling constants and porforming a conformal transformation) to find a metric gμν that locally reduces to the Minkowski flat spacetime metric ημν. ... the geodesics of gμν, are the trajectories of free falling bodies.

As for the SEP, it additionally enforces: the extension of the validity of WEP to self-gravitating bodies; and the validity of the EEP (that is, the LLI and LPI, of course), to local gravitational experiments. The only known theory that satisfies the SEP is General Relativity."

"The definition of ‘test particles’, i.e., how small must a particle be so that we can neglect its gravitational field, (note that the answer will probably be theory-dependent)"

"Thorne and Will proposed the following metric postulates:

1. Defined in space-time, there is, a second rank non-degenerate tensor, called a metric, gμν ;

2. If Tμν is the stress-energy tensor, associated with non-gravitational mater fields, and if ∇μ is a covariant derivative derived from the Levi Civita connection associated with the metric above, then ∇μT μν = 0 .

Theories that satisfy the metric postulates are called metric theories. We note two things about the metric postulates: first, that geodesic motion can be derived from the second metric postulate [3, 17]; second, that the definition of T μν is somewhat vague and imprecise, as is the notion of non-gravitational fields [18]."

"Mach’s principle, (which we will discuss briefly later) a sort of philosophical conjecture about inertia and the matter distribution of the Universe; the principle of equivalence; the principle of covariance; the principle of minimal gravitational coupling, which is stated as saying that “no terms explicitly containing the curvature tensor should be added in making the transition from the special to the general theory”; and, the correspondence principle, that in the case of gravitation means that GR must, in the limit of weak gravitational fields reduce to Newton’s gravitation."

"In the context of theories of gravitation, as was already stated, spacetime is a 4-dimensional manifold, where we define a symmetric non-degenerate metric gμν, and a quantity related to parallel transport called a connection (see [19] for an introduction, and [20] for a more advanced treatment), Γλμν. This connection, by its relation with parallel transport leads naturally to a definition of derivative adapted to curved manifolds, this is the covariant derivative denoted ∇ ? , in general. It’s definition is:

∇μTνσ =∂μTνσ +ΓνμαTασ −ΓαμσTνα. (2.2)

It is important to note, that we have made no association of the connection Γνμν with the metric gμν. This will be an extra assumption and, there will be a connection related to the metric called the Levi-Civita connection. We will use the symbol ∇ ? to denote this general covariant derivative, and ∇ denotes the one obtained from the Levi-Civita connection ... The notion of curvature of a manifold is given by the Riemann Tensor, which can be constructed from this generic connection as follows:

R^μνσλ = −∂λΓ^μνσ + ∂σΓ^μνλ + ΓμασΓανλ − Γ^μαλ Γ^ανσ ,which does not depend on the metric and is antisymmetric in it’s last indices. To describe the relation between the connection and the metric, we introduce the non-metricity tensor: Qμνλ =−∇ ?μgνλ (2.4)

and, the Weyl vector:

Qμ = (1/4)Q ν^μ^ν (2.5)

which is just the trace of the non-metricity tensor in it’s last two indices. Moreover, the antisymmetric part of the connection is the Cartan Torsion Tensor:

Sμνλ = Γλ[μν] ."

"One of the traces of the Riemann tensor is called the Ricci Tensor. Now, there are two possibilities for this contraction, either the first and the second indices or, the first and the third are contracted (due to the antisymmetry of the Riemann tensor, a contraction of the first and the fourth indices is equal to a contraction of the first and the third, with an additional minus sign), ... Rμν ≡ Rσμσν = −Rσμνσ or, We will thus obtain that the second tensor R ′ will be the antisymmetric

R′μν ≡ Rσσμν .(2.7) part of the first Rμν for a symmetric connection. The tensor quantity Rμν

is, of course, nothing but the usual Ricci tensor: Rμν = Rλμλν =∂λΓλμν −∂νΓλμλ +ΓλσλΓσμν −ΓλσνΓσμλ (2.8)

R′μν = −∂νΓααμ+∂μΓααν"

"Using the metric, — up to now the tensors have been independent from it —, to contract Rμν , we may obtain the usual Ricci scalar, whereas through the use of R′μν we get a null tensor, since the metric is symmetric and R′μν is antisymmetric. That is:

R = gμνRμν and, gμνR′μν ≡ 0. (2.10)"

"The principles stated above, are too general if we want to restrict ourselves only to GR. If we are to obtain Einstein’s theory, we will have to make further assumptions. This section will explore these assumptions.

We state them here briefly for future reference.

1. Torsion does not play any fundamental role in GR: Sμνλ = 0;

2. The metric is covariantly conserved: Qμνλ = 0;

3. Gravity is associated with a second rank tensor field, the metric, and no other fields are involved in the interaction;

4. The field equations should be second order partial differential equations;

5. The field equations should be covariant."

"One of the features of the above discussion was the independence of the connection relative to the metric, this was an attempt to get a general set of characteristics that a theory must obey. However, the second metric postulate calls upon a notion of covariant derivative — and consequently of connection —, that is linked to the metric. This choice of connection — the Levi-Civita connection —, is one of the most fundamental assumptions of GR. To fulfill this, it turns out that we need two things: firstly, the symmetry of the connection with respect to it’s two lower indices, that is:

Γαμν =Γανμ ⇔ Sλμν =0. (2.11) Secondly, the metric must be conserved by the covariant derivative — or,

covariantly conserved: ∇ ?λgμν =0 ⇔ Qλμν =0. (2.12)

The assumption (2.11) means that space-time is torsionless, while (2.12) implies that the non-metricity is null. With these choices, the connection takes the Levi-Civita form

{^αμν} = (1/2)g^α^β(∂μgνβ + ∂νgμβ − ∂βgμν) (2.13)"

"In Newtonian Gravity, the equation that describes the dynamics of the gravitational potential, is Poisson’s equation, ∇2φ = 4πρ. Einstein, in his original derivation of the field equations of GR, relied on a close analogy with this equation. In fact, the equations of GR in empty space are simply Rμν = 0, where this Ricci tensor has been constructed not from the most general connection but, out of the Levi-Civita affinity (2.13). This is in good analogy with Laplace’s equation ∇2φ = 0, since the Ricci tensor is a second order differential expression on the components of the connection."

"However, to extend this analogy to the case where we have matter, some extra assumptions must be made. The choice of the field(s) is the first assumption: in GR the only field of the theory (the only one whose dynamics we want to describe), is the metric. All other fields, are considered ‘matter fields’, i.e., sources of the ‘gravitational field’. Therefore, we impose (only in GR) that gravity is associated to no field other than the second rank tensor field that represents the metric. This also means of course, that the field equations should have a left side depending only on the metric and, a right side containing the dependence on all other fields, the ‘matter fields’. As it will turn out, the object that generalizes the distribution of the ‘matter’, and hence plays the role of the source of the field is, the Stress-Energy Tensor Tμν see [21] for a detailed discussion, and [3, 18] for some problems related to the definition of this quantity.

Second, if we are to have an analogy with Poisson’s equation, then our field equation must be a second order differential equation. As for the last requirement above, it stems from the second point in the Dicke framework, mentioned in section (2.1).

With these assumptions, if we follow the original derivation by Einstein see [15, 21, 22], we will obtain the field equations for GR:

Gμν = Rμν − (1/2)gμν = kTμν . (2.14)"

*"Wormhole" generalizes to Star Gate Time Travel Machine that is also an effective teleporter for large objects. This is not the same as "quantum teleportation." Do not confuse the two. - wrote Jack Sarfatti*

"Some of the exact solutions to Einstein’s equations are so important, that they merit investigations about whether or not they are still solutions to a general MTG. That is to say, for example: we know of the existence of a static, spherically symmetric, (possibly asymptotically flat), solution to Einstein’s equations, but is there such a solution (and, is it the same one) in, say, scalar- tensor theories or, in f(R) modifications of gravity? (For just on example see [25].) Similarly, are there any spatially homogeneous and isotropic, constant curvature solutions to a generic MTG? The two solutions just mentioned are, of course, Schwarzschild’s solution and Friedman-Lemaitre-Robertson-Walker type solutions respectively.

We devote this section, therefore, to a brief exposition of these two solutions, along with a third type of exact solution called wormholes. This is a rather different type of solution, since no known astrophysical objects are described by this type of solution, that is, there are no wormholes that we know of. Wormholes are just a theoretician’s probe of the foundations of a gravitational theory, they are “gedanken-experiments”, and we consider them as such."

*Jack Sarfatti did write: The flying saucers are coming through real wormholes or Star Gates" like in the sci-fi TV series.* *Don't believe the naysayers. Damn their torpedos full warp ahead. Check out the thriller "The Star Gate Conspiracy" by Picknett and Prince.*

"A wormhole solution is characterized by the following space-time metric:

ds^2=e^2Φ(r)dt^2 -(1−b(r)/r)^-1dr^2 -r^2(dθ^2 +sin^2θdφ^2) , (2.69)

where Φ(r) and, b(r), are arbitrary functions of the coordinate r. The function Φ(r), is related to the gravitational redshift and, is therefore called the redshift function; as for b(r), it is termed the shape function, since it determines the shape of the wormhole throat, as will be shortly seen."

"There are also a set of conditions, aimed at insuring that a traveler could actually use a wormhole. These are called traversability conditions, see [31, 32], for a detailed account. We refer explicitly, two of these conditions: one is linked with the gravitational acceleration felt by an observer at the initial and final points of his journey, this is, g = −(1 − b/r)^−1/2Φ′ ? −Φ′, and should be less the or equal to earth’s acceleration so that the condition |Φ′| ≤ g⊕, must be met; the other, is related to the redshift of a signal sent from the initial or final point, towards infinity, this is, ?λ/λ = e^−Φ −1 ≈ −Φ so that, we must have |Φ| ? 1. Given the first condition, a usual choice is to have the redshift function constant Φ′ = 0."

"Recent developments in observational cosmology brought about the need for two different phases of accelerated expansion of the universe. The first is inflation, that supposedly occurred in the early stages of the universe, and was succeeded by a radiation dominated expansion era. The second phase is the late-time cosmic acceleration, that is occurring in our present era. For the reasons we stated above, in section, 2.6.2, (equation 2.56), there is an intrinsic difficulty in the description of accelerated expansion since the condition for it’s occurrence in FLRW models is ρ + 3p < 0, and therefore if we use a fluid with an equation of state ω = p/ρ, this condition becomes ω < −1/3, which in turn implies the use of a negative pressure fluid.

There is a ‘natural’ choice for the source of this acceleration. The cosmological constant Λ, corresponding to a ωΛ = −1. The introduction of this constant, leads to a period of accelerated expansion."

"This interpretation however, suffers from serious drawbacks since this cosmological constant can not be easily (if at all), interpreted as a vacuum density in the context of field theories. The Λ term, in equation (3.2), is the source of another problem: once the possibility of a non-null cosmological constant has been introduced, setting this term (near to) to zero needs to be justified, just as setting any other term in any other equation to null. One is thus led to the so called ‘cosmological constant problem’ [34].."

* Jack Sarfatti comment: However, this problem is solved in my future hologram model discussed in my Journal of Cosmology paper Vol 14, April 2011 on line. There is also a natural explanation for the Arrow of Time and why we age as the universe accelerates speeding up its expansion rate.*

"Another possibility, for the source of these accelerated expansion periods, comes from scalar fields φ, with slowly varying potentials. These have been extensively studied and many variations of this theme exist — Quintessence, K-essence, Tachyon Fields, models using the Chaplygin gas, to name but a few [30]. However, no particular choice of field and potential, seems to generate a model in perfect accord with the experimental data. A particularly thorny problem appears to be, the possibility that the dark energy equation of state “crosses the phantom divide” (see [30], sec. V-D), that is, it may be in the region ωφ < −1."

"Wormholes — as mentioned in section 2.6.3 — are hypothetical tunnels in spacetime, possibly through which observers may freely traverse. However, it is important to emphasize that these solutions are primarily useful as “gedanken-experiments” and as a theoretician’s probe of the foundations of general relativity. In classical general relativity, wormholes are supported by exotic matter, which involves a stress-energy tensor that violates the null energy condition (NEC) [32, 33]. Note that the NEC is given by Tμνkμkν ≥ 0, (Eq. 2.61), where kμ is any null vector. Thus, it is an important and intriguing challenge in wormhole physics to find a realistic matter source that will support these exotic spacetimes."

*Jack Sarfatti comment: Virtual bosons anti-gravitate as dark energy. Virtual fermion-antifermion pairs (closed loop spinor Feynman diagrams) gravitate as dark energy - at least in free quantum field theory combined with the equivalence principle (SEP) and local Lorentz invariance.*