Let's go back to explore more of the multi-verse concept as it arises from M-theory. For convenience we include the same image of the topological 5-dimensional form from the previous blog. Prior to doing so, a brief review of basic Brane theory. First, to better familiarize casual readers with the distinction between particles, strings and branes, Fig. 1 is useful.
In the diagram, one begins with a zero-dimensional point particle. Let the particle move, say from point x1 to point x2 and it sweeps out a linear dimensional ‘string'. Take the 1-dimensional string and move it through space and it sweeps out a 2-dimensional band or what is called a p-brane. Take this p–brane and sweep it out in a perpendicular direction and one obtains a higher (p +1) dimensional world volume.
The quantitative description of brane phenomena is enormously complex and far beyond the scope of this blog, or even a small book. However, some simple insight and appreciation for the complexity can be gained by looking at the simplest brane of all, the D-0 brane consistent with the point particle. Without going into the details of obtaining the (fiber) bundle map, we require for completeness the tachyon field described by the expression for T(x) given in the upper right of Fig. 2, where the summation is taken from u= 1 to u = 9, with x_u the coordinates for the rotation space (group) R(9), and Gamma_u , the matrices needed. Then the tachyon field consistent with the above would be given by the small matrix expression in the upper right of Fig. 2.
For a system of D0- branes (see definition in last blog) parameterized by such matrices and defined by the given tachyon field (T(x)), we have a system of 9 x 9 matrices, defined by the large array shown in Fig. 2. Since the fields, and dynamics (as well as the tachyon field) are occurring at the level of the vacuum, that is the first place to start. Thus, to manipulate the D-brane in Fig. 2 it seems clear that energy consonant with a vacuum energy – or more exactly, a vacuum energy density- is needed. At the level of the vacuum, the smallest scale of fluctuations are at the level of the Planck Length, L (P)= ( h-bar G/ c^ 3 )^1/2 = 1.6 x 10 ^-33 cm, where h-bar again, is h/2 pi.
The energy density attendant on vacuum manipulation must have some lower limit. We use:
rho = {E/ c ^2}/ (L( P) )^3 = [ h-bar c/ L (P) )/ c ^2 ] / (L(P) )^3 ~ 10 ^94 g/cm^3
This is an astounding amount of mass-energy density! To give a rough comparison, the mass of the Sun is 2 x 10^33 g. The mass of the Milky Way galaxy is roughly 10 ^45 g. The mass of 100 billion galaxies the size and mass of the Milky Way is ~ 10^56 g. In other words, the mass density or energy density shown is more than ample to account for an entire universe!
The pure energy equally vast. More extraordinarily, the mean density of ordinary matter in the universe is: 4 x 10 ^-31 g/cm 3 , which means the vacuum mass-energy density is a factor ~ 10 ^125 greater!
We reiterate here that:
1) D-branes, by definition, are dynamical objects that can fluctuate.
2) The dynamical aspect is a result of the ability of D-branes to couple to closed strings (Fig. 2) Recall here that the latter embody a description of gravity and M-theory proper combines supergravity with Brane theory.
Now, we look at the putative multi-verse as shown in the bottom diagram.
Each longitude circle is a quotient space q1 and each latitude circle is a quotient space q2. More exactly, q2 denotes the fifth dimensional connector that links multiple parallel universes. Each of these can be represented by one single longitudinal loop marking off a coordinate on q1 (e.g. Θ). Since time can be broken down in fundamental units of tau (τ) then each q1 is separated by its neighbor universe by one tau. This neighbor is impossible to access from within the fixed coordinate quotient space (q1) but can be accessed if one can find a way to get to q2 and traverse it. Thus, in simplest terms, each “slice” q1 can be regarded as an alternate or parallel universe.
More explicitly, one may adopt absolute toroidal (Θ,φ) coordinates to locate events in the multiverse system. There are a number of fascinating aspects of this model. First, the toroidal hypersphere implies an endlessly repeating universe that is the same in each new cycle[1] - instead of being different (i.e. with different physical constant as the reprocessed model demands). The key to this aspect is the fact that the radius R remains constant.
A second intriguing aspect is that the exact same point of space-time occurs for each "beginning" (Big Bang) and "end" ("Big Crunch"). This point is easily identified in the diagram as the most constricted part of the interior "hour-glass" shape defining the inner wall of the torus-hypersphere. Third, the overall configuration is exactly the same as that defined for the high-energy "twistors" used in the 1980's by Roger Penrose and his colleagues at Oxford.[2] The twistors are discrete points of space, each defined by 4 real numbers and 4 imaginary numbers.
The important geometric concept here, discerned from the situation shown, is that two distinct parallel universes (recall these are given by different longitudinal coordinates, Θ) are linked by the bundle’s vortex. Thus, such a linkage between parallel universes (within the defined multi-verse) is equivalent to an enormous short cut for the craft that seeks to exploit it. What would the conditions have to be for linkage? Let the two parallel universes be distinguished by a 1-τ difference in fundamental time parameter, viz. [1 + 2τ] and [1 + 3τ], then we would require for connection, a mapping such that:
f:X -> X = f(Θ,φ) = (Θ, 2φ)
f:X -> X = f(Θ,φ) = (Θ, 3φ)
which means the absolute coordinate φ is mapped onto itself 2 times for [Universe A] and mapped onto itself 3 times for [Universe B]. Clearly, there’ll be coincidences for which: f(Θ,2φ) = f(Θ,3φ) wherein the two universes will 'interweave' a number of times. For example, such interweaving will occur when φ = π/2 in [A] and φ = π/3 in [B]. The total set or system of multiple points obtained in this way is called a Synchronous temporal matrix. The distinguishing feature of this matrix is that once a single point is encountered, it is probable that others will as well. If one hyperspace transformation can occur linking parallel universes, A and B, then conceivably more such transformations can occur, linking A and C, D and E etc.
What if both absolute toroidal coordinates (Θ,φ) map into themselves the same number of times? Say, something like:
f:X -> = f(Θ, φ) = (2Θ, 2φ): Universe A
f:X -> = f(Θ, φ) = (3Θ, 3φ): Universe B
For example, given the previous conditions for coordinate φ, now let 2Θ = 3Θ for discrete values of Θ (e.g. 2π). For all multiples of 2π, the same toroidal cosmos will be experienced - if the absolute time coordinates are equal (e.g. π/2 = φ in A, and π/3 = φ in B) then we will have: Universe A = Universe B. The exact same physical state-space prevails in both A and B. For all intents and purposes there is no difference between them. In this case, we say that there exists an interpenetration of different parallel universes. Note that though the physical state spaces (e.g. with constants h, G, e/m, etc. )may be alike, they can still differ in dimensionality. And we cannot disregard fractal dimensionality.
IF one has this condition, THEN it is feasible that the (David) Deutsch experiment to detect the interphasing of a parallel universe (see link in previous blog) can be carried out, and the penetration of our universe by a parallel one validated. If the topological condition above has not been met, then we expect the Desutsch experiment will render a negative result, but this doesn't mean the parallel universe theory is necessarily invalid- only that the specific topological condition hasn't been met! (Absence of evidence here is not evidence of absence).
[1] This could also be interpreted as a "timeless" universe. See, e.g. Rucker, R.: 1977, Geometry, Relativity and the Fourth Dimension, Dover Publications, New York, p. 105. According to Rucker:"There is no last time around or next time around because nothing is moving...we feel that we are going through time but that is an illusion."
[2] For an excellent article on twistors, see: Spinning New Realities, by R.L. Forward, in Science 80 (December, 1980), p. 40. The numbers defining them have the form: a + bi, where a is the real part and bi is imaginary since i = [(-1)]^1/2. The complete a + bi is known as a complex number. Readers can refer back to the succession of blogs I did on complex numbers, starting with: http://brane-space.blogspot.com/2010/03/looking-at-complex-numbers.html
In the diagram, one begins with a zero-dimensional point particle. Let the particle move, say from point x1 to point x2 and it sweeps out a linear dimensional ‘string'. Take the 1-dimensional string and move it through space and it sweeps out a 2-dimensional band or what is called a p-brane. Take this p–brane and sweep it out in a perpendicular direction and one obtains a higher (p +1) dimensional world volume.
The quantitative description of brane phenomena is enormously complex and far beyond the scope of this blog, or even a small book. However, some simple insight and appreciation for the complexity can be gained by looking at the simplest brane of all, the D-0 brane consistent with the point particle. Without going into the details of obtaining the (fiber) bundle map, we require for completeness the tachyon field described by the expression for T(x) given in the upper right of Fig. 2, where the summation is taken from u= 1 to u = 9, with x_u the coordinates for the rotation space (group) R(9), and Gamma_u , the matrices needed. Then the tachyon field consistent with the above would be given by the small matrix expression in the upper right of Fig. 2.
For a system of D0- branes (see definition in last blog) parameterized by such matrices and defined by the given tachyon field (T(x)), we have a system of 9 x 9 matrices, defined by the large array shown in Fig. 2. Since the fields, and dynamics (as well as the tachyon field) are occurring at the level of the vacuum, that is the first place to start. Thus, to manipulate the D-brane in Fig. 2 it seems clear that energy consonant with a vacuum energy – or more exactly, a vacuum energy density- is needed. At the level of the vacuum, the smallest scale of fluctuations are at the level of the Planck Length, L (P)= ( h-bar G/ c^ 3 )^1/2 = 1.6 x 10 ^-33 cm, where h-bar again, is h/2 pi.
The energy density attendant on vacuum manipulation must have some lower limit. We use:
rho = {E/ c ^2}/ (L( P) )^3 = [ h-bar c/ L (P) )/ c ^2 ] / (L(P) )^3 ~ 10 ^94 g/cm^3
This is an astounding amount of mass-energy density! To give a rough comparison, the mass of the Sun is 2 x 10^33 g. The mass of the Milky Way galaxy is roughly 10 ^45 g. The mass of 100 billion galaxies the size and mass of the Milky Way is ~ 10^56 g. In other words, the mass density or energy density shown is more than ample to account for an entire universe!
The pure energy equally vast. More extraordinarily, the mean density of ordinary matter in the universe is: 4 x 10 ^-31 g/cm 3 , which means the vacuum mass-energy density is a factor ~ 10 ^125 greater!
We reiterate here that:
1) D-branes, by definition, are dynamical objects that can fluctuate.
2) The dynamical aspect is a result of the ability of D-branes to couple to closed strings (Fig. 2) Recall here that the latter embody a description of gravity and M-theory proper combines supergravity with Brane theory.
Now, we look at the putative multi-verse as shown in the bottom diagram.
Each longitude circle is a quotient space q1 and each latitude circle is a quotient space q2. More exactly, q2 denotes the fifth dimensional connector that links multiple parallel universes. Each of these can be represented by one single longitudinal loop marking off a coordinate on q1 (e.g. Θ). Since time can be broken down in fundamental units of tau (τ) then each q1 is separated by its neighbor universe by one tau. This neighbor is impossible to access from within the fixed coordinate quotient space (q1) but can be accessed if one can find a way to get to q2 and traverse it. Thus, in simplest terms, each “slice” q1 can be regarded as an alternate or parallel universe.
More explicitly, one may adopt absolute toroidal (Θ,φ) coordinates to locate events in the multiverse system. There are a number of fascinating aspects of this model. First, the toroidal hypersphere implies an endlessly repeating universe that is the same in each new cycle[1] - instead of being different (i.e. with different physical constant as the reprocessed model demands). The key to this aspect is the fact that the radius R remains constant.
A second intriguing aspect is that the exact same point of space-time occurs for each "beginning" (Big Bang) and "end" ("Big Crunch"). This point is easily identified in the diagram as the most constricted part of the interior "hour-glass" shape defining the inner wall of the torus-hypersphere. Third, the overall configuration is exactly the same as that defined for the high-energy "twistors" used in the 1980's by Roger Penrose and his colleagues at Oxford.[2] The twistors are discrete points of space, each defined by 4 real numbers and 4 imaginary numbers.
The important geometric concept here, discerned from the situation shown, is that two distinct parallel universes (recall these are given by different longitudinal coordinates, Θ) are linked by the bundle’s vortex. Thus, such a linkage between parallel universes (within the defined multi-verse) is equivalent to an enormous short cut for the craft that seeks to exploit it. What would the conditions have to be for linkage? Let the two parallel universes be distinguished by a 1-τ difference in fundamental time parameter, viz. [1 + 2τ] and [1 + 3τ], then we would require for connection, a mapping such that:
f:X -> X = f(Θ,φ) = (Θ, 2φ)
f:X -> X = f(Θ,φ) = (Θ, 3φ)
which means the absolute coordinate φ is mapped onto itself 2 times for [Universe A] and mapped onto itself 3 times for [Universe B]. Clearly, there’ll be coincidences for which: f(Θ,2φ) = f(Θ,3φ) wherein the two universes will 'interweave' a number of times. For example, such interweaving will occur when φ = π/2 in [A] and φ = π/3 in [B]. The total set or system of multiple points obtained in this way is called a Synchronous temporal matrix. The distinguishing feature of this matrix is that once a single point is encountered, it is probable that others will as well. If one hyperspace transformation can occur linking parallel universes, A and B, then conceivably more such transformations can occur, linking A and C, D and E etc.
What if both absolute toroidal coordinates (Θ,φ) map into themselves the same number of times? Say, something like:
f:X -> = f(Θ, φ) = (2Θ, 2φ): Universe A
f:X -> = f(Θ, φ) = (3Θ, 3φ): Universe B
For example, given the previous conditions for coordinate φ, now let 2Θ = 3Θ for discrete values of Θ (e.g. 2π). For all multiples of 2π, the same toroidal cosmos will be experienced - if the absolute time coordinates are equal (e.g. π/2 = φ in A, and π/3 = φ in B) then we will have: Universe A = Universe B. The exact same physical state-space prevails in both A and B. For all intents and purposes there is no difference between them. In this case, we say that there exists an interpenetration of different parallel universes. Note that though the physical state spaces (e.g. with constants h, G, e/m, etc. )may be alike, they can still differ in dimensionality. And we cannot disregard fractal dimensionality.
IF one has this condition, THEN it is feasible that the (David) Deutsch experiment to detect the interphasing of a parallel universe (see link in previous blog) can be carried out, and the penetration of our universe by a parallel one validated. If the topological condition above has not been met, then we expect the Desutsch experiment will render a negative result, but this doesn't mean the parallel universe theory is necessarily invalid- only that the specific topological condition hasn't been met! (Absence of evidence here is not evidence of absence).
[1] This could also be interpreted as a "timeless" universe. See, e.g. Rucker, R.: 1977, Geometry, Relativity and the Fourth Dimension, Dover Publications, New York, p. 105. According to Rucker:"There is no last time around or next time around because nothing is moving...we feel that we are going through time but that is an illusion."
[2] For an excellent article on twistors, see: Spinning New Realities, by R.L. Forward, in Science 80 (December, 1980), p. 40. The numbers defining them have the form: a + bi, where a is the real part and bi is imaginary since i = [(-1)]^1/2. The complete a + bi is known as a complex number. Readers can refer back to the succession of blogs I did on complex numbers, starting with: http://brane-space.blogspot.com/2010/03/looking-at-complex-numbers.html
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