On a Final Theory of Mathematics and Physics by Michael Rios

The determinant is a measure of the action. A state |Ψ> = A_{ab}|abc> for the indices a = {0, 1} transforms as (2, 2) under SL(2, K)_a x SL(2, K)_b. We compute the determinant of this C_{ab} accordin to the FTS. The action is then S = pi det(A), where we can identify this matrix with the J3(O) 3x3 matrix.

The cubic equation

-det(A - λI) = λ^3 - tr(A)λ^2 + σ(A)λ - det(A)

where σ(A) = R(X, X') for the curvature of the 3 dim manifold of the eigenvalues. This was something I illustrate with the Morse index approach.

We may also see this in this three dimensional space as a volume form C_{abc} = V_a/\V_b/\V_c, (/\ = wedge) where the value of this is the determinant of this 3-way determinant. This is then in some way equivalent to the hyperdeterminant. This appears like something to explore. The value of these eigenvalues is given by a discrete set of points that define a polytope. This polytope defines the G_{abcd}. Again the action is defined by the determinant, or I think in general a Hessian. I think this connection to the polytope, something similar to a Kirwan polytope, and the SLOCC type is something interesting to work on.

LC

A diversion back to the subject of LQG:

The papers by Smolin http://arxiv.org/pdf/hep-th/0006137.pdf and http://arxiv.org/pdf/hep-th/0104050v1.pdf do make the point there is a duality between strings and loops. The duality largely focuses on SO(9,1) and supergravity. Straight up LQG is formulated in four dimensions. The thing that occurred to me is that maybe the LQG does not "turn on" at low energy. In other words low energy physics with all the CY manifolds compactified on a small scale is where there is no "loopy physics," but at higher energy in supergravity maybe these appear. This might then mean the physical states of the QFT as a quotient SO(4,2)/SO(4,1) is such that loop variables are removed from the physics at "large N" q-gravity.

I started to look at Carroll's paper "Consistency Conditions for an AdS/MERA Correspondence" http://arxiv.org/pdf/1504.06632v1.pdf that makes use of something I proposed in an FQXi essay a number of years ago

http://fqxi.org/data/essay-contest-files/Crowell_time_scaling.pdf?phpMyAdmin=0c371ccdae9b5ff3071bae814fb4f9e9

that AdS spacetimes have a renormalization group flow. This suggests a way of removing divergences from the ground state in physics. Physically it might suggest that the disappearance of quantum gravity as a "common sense quantization of general relativity" might be a reflection of this and why it is that the cosmological constant is so incredibly smaller than what would be ordinarily thought.

Cheers LC

Wow that is a 100 page paper. I am still digesting the Kalosh et all paper.

This seems to lead in some ways to what Connes recently published. I think that this all looks a bit like a "phase" of quantum states, similar to topological transitions of edge states in quantum Hall/Mott insulator states. Physics then looks very different at low energy, or for measurements along a long baseline, but different for high transverse momenta.

LC

Michael,

As I read and research deeper into this the more questions tend to occur. The paper by Ferrara, Gimon and Kallosh has been most interesting to study. I like the Hamiltonian dynamics approach they take. I also see this might connect to the Hessian methods I outlined earlier It appears that the entropy functions I(z) (z = (p, q) in pseudocomplex notation) is such that

I(z + δz) = I(z) + J(I)(z)δz + δz^†H(I)(z)δz.

For J(I)(z) = Jacobian matrix and H(z) the Hessian matrix with second derivatives. Clearly H(I)(z) = J(∇I)(z). The near horizon moduli can then be

q_ОЈ + (∂I(z)/∂p)_ОЈ + X(∂^I(z)/∂q∂p)_ОЈ = 2iZ(О")M_ОЈ

p^ОЈ + (∂I(z)/∂q)^ОЈ + X(∂^I(z)/∂q∂p)^ОЈ = 2iZ(О")L^ОЈ

where the X is meant to indicate twistor geometry with

П‰'^A = П‰^A + X^{AA'}ПЂ_A'

In effect the inclusion of the second derivative "twistorizes" this.

This seems to point to an interesting question. The SLOCC correspondence with black holes, BHQC (black hole qubit correspondence) involves state that are cosets for N = 2 and 8 SUSY with ВЅ, Вј and 1/8 BPS, or nonBPS as well. The cosets are E8(-24}/E7(-25)xSL(2,R) for d = 3 and E7(-25)/E6(-78)xU(1). For N = 4 SUSY we have states that are conformal field theories equivalent to SO(4,2)/SO(4,1) ~ AdS_5, and the isometry group for these is SO(4,2) ~ SU(2,2). This leads to twistor space, and it seems that there is something subtle going on here. As I indicated the transition to quaternions leads to some sort of generalization of twistor geometry. The collineations e6 correspond to isometries of OP^2 by F4 as SO(9,1) and SO(9). The points that are collinear in OP^2 correspond in C^3 to SL(4,C). These are points collinear in C^3, and this complex conformal group defines the set of lines connected to a loop. The generalization is to SL(4,H) and the Grassmannian is

G_{4,4} = SU(4,4)/(SU(4)xSU(4))

That embeds T-space. There is something going on with N = 4 SUSY with conformal field theoretic formalism and coset state space = CFT_4 ~ AdS_5 and N = 2 or 8 SUGRA and magic SUGRA.

Happy weekend LC

The connection between SL(4,C) and CP^3 is seen with

h_5(C) ~= h_4(C)вЉ•C^4вЉ•R

That is a 2x2 matrix of 4-component objects, spinors П€ and vectors П† in C^4 and h_4(C). The Jordan algebra h_4(C) ~ CP^3 and in conjuction with the C^4 tells us that we have a CP^{3|4} superspace for twistor theory.

It would appear best at first to consider

h_3(H) ~= h_2(H)вЉ•H^2вЉ•R.

This is just a way of twistor theory based on H^2 is that H^2 ~ C^4, and of course we have a different meaning to projective spaces. The standard theory has T = C^4 with projective twistor space CP^3 = P(C^4) < --- F_12(T) --- > G_{2,4}(C). The G_{2,4}(C) = U(4)/U(2)^2, or in signatured spacetime SU(2,2)/SU(2)xSU(1,1). The quotient between SU(2,2) = SO(4,2) and SU(2)xSU(1,1) ~ SO(3,1) is the Grassmannian space of this space of 2-planes in 4-space. The quotient SO(4,2)/SO(4,1) is the AdS_5 or moduli space for the quaternion bundle. This has a relationship to G_{2,4}(C) as its quotient is with SO(3,1) вЉ‚ SO(4,1). The CP^3 is projective twistor space PT and the manifold F_{12}(T) is a five dimensional space that through this double fibration maps information in this space to CP^3.

This change is a bit similar to considering the Dirac equation according to the Dirac equation that is a 4-column made of complex objects. We can of course consider this equivalently as a a pair of complex fields that define a quaterionion. This is a bit like working with the Dirac equation directly or with a pair of Weyl equations that define the same thing. Twistor theory works in exactly this construction.

What I propose is that we consider

h_5(H) ~= h_4(H)вЉ•H^4вЉ•R.

The group SL(4,H) constains SL(4,C)xSL(4,C) and we may then embed two copies of twistor space in here. We have instead of conjugation *i --- > -i there is in addition so(n) algebras for the elements with z = x + iu + jv + ky, from which we can build up Clifford algebras. The Clifford algebras correspond to a gauge theory. I illustrated in an attachment how quaternions give rise to gauge theory. In this way the theory obeys the symmetry of a Clifford algebra corresponding to the symmetry of the gauge fields.

LC

The H_1 and H_2 I presume as elements of the J3(O) then I presume that X is then some nonassociating element The equation

dX = [H_1, X, H_2]

corresponds to the associator. The odd think is that the Jacobi identity for

6[x, y, z] = [x, [y, z]] + [y, [z, x]] + [z, [x, y]],

which in standard group theory is zero. This is probably one reason that most people reject the idea of nonassociative mathematics. I myself would be tempted to say that for the real eigenvalues of J3(O) diagonalized with F4 that the associator has zero contribution.

The extension to E6 = SL(3, O) involves the subgroups sl(2,O) and SU(2, O). We the include the real forms of D5 and B4 in addition to the eigenvectors of F4 that are the Hurwitz quaternions. This means cranking out the roots and weights of PSO(9,1) and SO(9). The SO(9) is contained in the F4, and so we now have to contend with SO(10) or SO(9,1). For the SO(9,1) we have Lorentz boosts now to contend with. The elements H_3в--¦X, the Jordan product then evaluates according to the work I did last month with

Tr(Aв--¦Q_О») = 2П‰([П‡, П‡'])

For Q = v_λ(v_λ)^†/\langle (v_λ)^†|v_λ\rangle

Which is a projector in J3(O). Now since the eigenvalues can be imaginary it poses I think no particular problem with the closure of the Jacobi identity.

That H3 is 26 dimensional does give a sort of triality situation. The 26 dimensions are "half" of the F4, where there are two dual root systems. The 24 roots in the 24 cell are (+/-ВЅ, +/-ВЅ, +/-ВЅ, +/-ВЅ) = 16 roots and (+/-1, 0, 0, 0) = 8 roots with units permuted. These Hurwitz quaternions are z = a + bi + cj + dk. There is a dual set of 24 roots (+/-1, +/-1, 0, 0). These fill out the 48 roots of the F4 with the additional 4 weights.

We have then the E6(-26) is a set of Hurwitz quaternions plus another "half" or fundamental representation of F4. The invariant S = ПЂ sqrt{I_1}is then cubic in the elements from the three fundamental representations. The quaternion roots when summed give three fundamental quantities associated with the action or invariant. It is then interesting to note that E7(-25) is then 133 - 25 which contain 108 dimensions, suggesting a span equivalent to two F4s.

Just to further the argument with respect to 5-branes, the generator of the partition function for the 5-brane (NS5-brane) is ПЂ sqrt(Q_1Q_5n_5) with n_5 = EL/2ПЂ for E the energy of the brane and L its scale. The charge Q_5 is the charge of the NS5-brane and Q_1 that of a D1 on the brane. The determinant system is of the same form. Given that the cubic system is for the d = 5 system the space in five dimensions is identified as the NS5-brane.

The invariant I_1(p, q) defines the metric with z = (p, q)

ds_4^2 = (1/I_1(z))(dt + ПЃ_idz^i) + I_1(z)dx_idx^i

where the entropy function I(z + δz) = I(z) + J(I)(z)δz + δz^†H(I)(z)δz. For J(I)(z) = Jacobian matrix and H(z) the Hessian matrix with second derivatives. Clearly H(I)(z) = J(∇I)(z). The near horizon moduli can then be

q_ОЈ + (∂I(z)/∂p)_ОЈ + X(∂^I(z)/∂q∂p)_ОЈ = 2iZ(О")M_ОЈ

p^ОЈ + (∂I(z)/∂q)^ОЈ + X(∂^I(z)/∂q∂p)^ОЈ = 2iZ(О")L^ОЈ

where the X is meant to indicate twistor geometry with

П‰'^A = П‰^A + X^{AA'}ПЂ_A'

In effect the inclusion of the second derivative "twistorizes" this. We then have the twistor connection ПЃ^i = Пѓ^i_{AA'}X^{AA'}. The holomorphic coordinate variable of interest is

t^ОЈ(p,q) = (M_ОЈ/M^0) =

(p^Σ + (∂I(z)/∂q)^Σ + X(∂^I(z)/∂q^2)^Σ)

(p_0 + (∂I(z)/∂q)_0 + X(∂^I(z)/∂q^2)_0).

For holomorphy we have ∂^I(z)/∂z^2 = 0 so that ∂^I(z)/∂q^2 = ∂^I(z)/∂p^2. The invariant then obeys

∂^I(z)/∂z∂z-bar = ∂^I(z)/∂q^2 + ∂^I(z)/∂p^2,

and the invarant is a Kahler potential.

The harmonic symplectic

H(z) = (H^a, H_a) = h + sum_s=1^nО"_s/|z - z_s|^p = h + О"/|z - z_s|^p

for the far field situation z >> z_s the invariant is I_1 ~= h + О"/z^p. The metric element is (∇xПЃ)_i = _i and ∇^2ПЃ_i = _i. The invariant is written as a product so that

(1/3!)c_{ijk}t^i t^j t^k = h + О"/z^p = e^{-2U(z)}.

The holomorphic coordinates corresponds to the NS5-brane a string on the brane and momentum associated with the energy scale of the brane.

In ten dimensions the line element determined by the 5 dimensions tangent to the branes, Greek indices, the directions tangent to the branes and transverse to the string (D1-brane) on the brane, Latin index n, and the spatial directions 1, 2, 3, 4 transverse to both the NS5-brane and the D1-brane, Lattice index j, and finally the index 5 in the direction the NS5-brane is moving,

ds^2 = A^{-1}[g_{ОјОЅ}dx^Ојdx^ОЅ + B(-dt^2 + dx_5^2)] + Adx^jdx^j + Cdx^ndx^n.

The event horizon in 10 dimensions is 8 dimensional. This has 5 dimensions tangent to the brane and 3 dimensions transverse to it. This then occurs in the metric term

A^{-1}r^2dО©^2.

The metric coefficient is ~ 1/r^p, for p = 5 - horizon dimension on the brane = 2. It is also dependent on the brane and string scales r_1 and r_5, from which we compute A^{-1}B. This is compared to the metric for I_1 above. These occur separately so that

A^{-1} ~ r_1^2r_5^2/r^2.

The area is then Area = r_1^2r_5^2. For the direction along the motion of the brane we have in Euclideanized metric the area is then

Area = r_1^2r_5^2n_5^2LV^4.

For L the scale of the brane and V^4 the volume in the 1, ..., 4 transverse directions. The distances are the square of the respective charges so that

Area = K sqrt{Q_1Q_5n_5}

This then connects S = ПЂ sqrt(det A^3) to the area of a black-brane or black hole.

LC

The Jordan product xâ--¦y = ½ (xy + yx) involves obvious Fermi-Dirac statistics as an anticommutator. The associator [x, y, z] = (xâ--¦y) â--¦z - xâ--¦(yâ--¦z) is computed as

[x, y, z] = ¼ ({x, y}z + z{x, y})

where if we employ {x, y} = g(x, y), the associator is then

[x, y, z] = g(x, y)z

Usually the anticommutator evaluates as a delta function, and so the associator with the Jordan product appears to be an operation with three elements that returns one element multiplied by this metric.

We might consider this in fermion space with V = C^N as the vector space for N particles, and V* the dual vector space. The vector space V contains elements of v = a^iv_i, for the index i = 1, ..., N and the dual space contains elements dual to these vectors, which can be thought of as 1-forms ω^i, so that v* = b_iω^i. In this way an elements of W = V⊕V* = (v, v*). The anticommutator is then {e_i, ω^j} = δ_i^j. We think of e_i as the same as a^† for the ith particle and ω^i as the annihilation operator a for the ith particle. From this of course any state my be built up as a sum of products of these operators e_i acting on the vacuum.

We are then able to look at the metric/anticommutator g(x, y) with x = a^ie_i + b_iω^i and y = c^ie_i + d_iω^i

g(x, y) = {x, y} = {a^ie_i + b_iω^i, c^je_j + d_jω^j} = a^id_jδ_j^i. + c^jb_iδ_i^j.

This then gives the associator

[x, y, z] = (a^id_i + c^ib_i)z.

The term a^id_i + c^ib_i = as a form of inner product. We may then of course extend this to V --- > V⊕R, where the states have a scalar and vector/form component to them as seen in Clifford algebras. This J(W) = V⊕R, J for Jordan product is such that for (x, a) and (y, b) then

(x, a)â--¦(y, b) = (ax + by, ab + )

For a change of signature on the second component, or equivalently (x, a) --- > (x, Ja) for J the pseudocomplex matrix, - ab is then the Lorentzian metric interval and vanishes on the light cone.

We may then consider what this means for the e6(-26). This has mcs f4 ⊕ R^26. The R^{26} is treated as the Lorentz space for the bosonic string. These 26 dimensions have one dimension for time, the direction of the string motion and 24 dimensions for the transverse polarizations of the string. The transverse directions correspond to the rank of the gauge group, which is then SO(24). The Regge trajectory α'M = k(N - 1), k = 1 or 4 for open and closed strings. We ignore the negative states as an unstable vacuum or tachyon and we then have the first massless states |Ω^{ij}> = a^ia^j|0>. The operator a^i is for -1. The antisymmetric part of |Ω^{ij}> corresponds to a gauge field, the antisymmetric part due to gravity, and the trace forms the dilaton and axion fields. The gauge field is SO(24), and the symmetric part contains independent 300 elements, which can form an SO(25), such that the traceless part is again SO(24), and the remaining 24 elements of the trace can form an SU(5) group. In fact the α'M = 2 states, a^i_{-3}|0>, a^i_{-1}a^j_{-2}|0> and a^ia^ja^k|0> give 24, 576 and 2600 and combine to give the 300 of SO(25).

This is not a physically reasonable theory, largely from experimental grounds. We then consider the replacement of the Virasoro algebra for the bosonic string with D = 26 anomaly cancellation with the superVirasoro algebra with the D = 10 anomaly cancellation term. In this case we now have a theory with 10 Dirac matrices. I of course am not going into the complexity of this, which is fairly extensive. With 10 Dirac operators we are now in the domain of SL(2, O) ~ SO(9, 1). Weyl-Majorana spinors or Î" in D = 10 have 16 components. With this we can now extract the SUGRA out of this theory of entanglement of states. It is recalled that this construction with E6(-26) and F4 does not pertain to field theory, but instead points to the coset construction of quantum states. The coset e6/f4 is ½ SUSY BPS BH, but the gauge theory and gravity emerges from this.

This connects back to the associator. In the context of the Jordan matrix this contains fermion theory. The above space W = V⊕V* = (v, v*) transforms by SL(2N, C). For N = 2 this is SL(4,C), for N = 4 this transforms by SL(8, C) ~ SL(4, N) and for N = 8 we have SL(4, O). SL(4, O) ~ SL(16, C) is the 8qubit SLOCC group. It appears that we can connect up SUGRA this way.

LC

I have another question on all of this, which might in some ways be an insight. The invariant of E6(6) is this cubic invariant. The matrix

J3 = q_{ab}П‰^{bc}q_{cd}П‰^{de}q_{ef}П‰^{fa}

is written with the 54 = 27⊕27, for 27 = 1⊕26, where these are sp(8). The action is then S = π sqrt{|J3|}. The determinant of this matrix is a 3-volume that has zero volume and if its bounding area ∂V = ∂J3/∂q = 0 ½ of supersymmetries are preserved. If the volume is zero and the area nonzero then ¼ SUSYs preserved and if both are nonzero then 1/8.

I have been thinking that this has a connection to the Weyl curvature and its action. Since this has gotten into conformal relativity this is a time where I can use this forum as a sound board to present some physics involving the Weyl curvature. Consider the Lagrangian

L = (1/2)sqrt{-g}C_{abcd}C^{abcd}

and the variation with respect to the metric g_{ij} is then

sqrt{-g}^{-1}ОґL/Оґg_{ij} = B^{ij} = 2C_{aijb}^{;ab} + C_{aijb}R^{ab} = 0.

This is comparatively simple for a conformally flat spacetime where R^{ab} = 0. An Einstein spacetime with R^{ab} = О»g^{ab} is also a solution to the equation. This is the Bach equation.

Suppose these two actions are the same. This then equates a quadratic function with a cubic function. The above functional differential with the metric is essentially the same as a derivative with respect to the 26 charges of the sp(8). The metric is then given by the 26 electromagnetic charges. This then gives action as an elliptic curve. This seems to be a physical way of looking at ideas of using Shimura and symmetric varieties in physics.

LC

To introduce more physical ideas into this subject, in particular magic SUGRA I have been musing over this for a while. I think it is important to keep in contact with physical ideas. I do at the end here consider the prospect that this connects with J3 and E6. This has to do with the general uncertainty principle and the tension on the brane of our universe. If the tension is zero then the fluctuations in the spacetime are tiny. However, if we are observing spacetime under a high magnification of a Heisenberg microscope then we are imposing high transverse momenta and the tension is huge. Under those conditions the fluctuations are quite large.

We might think of the situation like this. Suppose the uncertainty principle is

О"pО"x = Д§ + F(x)О"xО"t.

This is a version of the uncertainty principle modified for string theory

О"pО"x = Д§ + ОјО"x^2,

where ОјО"x = F(x)О"x/c = F(x)О"t. The constant Ој has been replaced with a force that depends upon distance or a region in space/spacetime. I have though a term F(x) that is written to indicate force. This leads to the spread in the width of the uncertainty

О"x = О"pc/2F(x) +/- c sqrt{О"p^2 - 4Д§F(x)/c}/2F(x)

From О"pО"x = Д§ + F(x)О"xО"t we can divide by О"t with О"x/О"t ~ c to get

F(x)О"x = О"pc - Д§/О"x

where the LHS is the work equivalent due to this force. We may write this according to О"W = О"pc - Д§/О"x, so that the work is seen to increase with О"p and decrease with О"x. This may also be expressed to connect the momentum-position uncertainty with a "quantum work-time uncertainty" so that

О"WО"t = О"pО"x - Д§.

For F = 0 then О"pО"x = Д§ and the LHS is zero. This illustrates how О"WО"t is associated with an increase in uncertainty.

If you think this looks similar to the entropy force of gravity results you are not too far off! The entropy force of gravity concerns the transverse displacement of a holographic screen. Here we are concerned with the stretching or longitudinal motion along the screen. The entropy force of gravity F = T∇S determines work as

О"W = FО"x = T∇SО"x = TО"S.

We employ E = (1/2)NkT which equals Mc^2, We then see that temperature is T = Mc^2/Nk. This is inserted into the equation for the entropy force

FО"x = (2Mc^2/Nk)О"S

The motion of the holographic screen by a distance О"x = Д§/mc results in

F = (Mmc^3/Nk Д§)О"S

and we use N = A/4L_p^2 = πR^2/(Għ/c^3). We then have that

F = 2GMm/(kR^2)О"S

Where we then assign the unit of entropy О"S = k/2 to get the result.

The NS5-brane has spacetime of four dimensions on a D-brane sandwiched between two other branes embedded in 5 dimensions. This sandwich is contained in an anti-de Sitter spacetime (AdS_5) of five dimensions. In four dimensions of spacetime a static gravity field in 3-dimensional space has the property that a two-dimensional surface bounding a mass will have the same number of line of flux through it independent of the radial size of the surface. Thus the force F multiplied by the surface area 4ПЂR^2 is a constant and equal to the gravity constant in 4-dimensions multiplied by the masses

GMm = 4ПЂFR^2,

Which leads to Newton's law of gravity

F = GMm/R^2

The 4ПЂ has been "absorbed into the definition of G. In five dimensions the surface bounding the mass is not two dimensional but three dimensional, which gives us the law

G_5Mm = 4ПЂFR^3

And so the force law in five dimensions is

F = G_5Mm/R^3.

The G_5 is the gravitational constant in five dimensions, and it reflects how the branes are "warped" in the embedding space by interacting with each other. G_5 in these units is Nm^3/kg^2 (N = Newton, m = meter, kg = kilogram). The relationship with the standard gravitational constant G with units Nm^2/kg^2 is that G_5 = GL, where L is a unit of length.

The brane tension, or equivalently the tension of strings attaching them together, is T = kL, for k = constant. The string length is L = 1/sqrt{R}, for R the curvature of the brane which we write as L = L_0^2/s, for L_0 the zero tension string with F(x) = 0 and s the "warp length," which is something that Randall worked out.

G_4 = (4ПЂ/3)О»G_5^2 = G_5/s

It is kind of neat. The big question I have is whether or not our four dimensional spacetime is not in fact on the conformal boundary of the AdS_5. This does go somewhat into the idea about changing gravity, which appeals to the physics of our brane in or on the AdS_5 and how gravity is related to the brane tension.

We also see that the tension depends upon the length L and for the tension at a minimum the Newton gravitational constant G_4 is small. For T or О» large the gravitational constant is huge. We then have a renormalization group flow for the gravitational constant. Also in addition G_4 = (Planck area)c^3/Д§, and for the tension large this is larger and the area of a Planck unit is larger. This is due to an impulse F(x)О"x = О"pc - Д§/О"x that "stretches" the horizon or Planck unit of area. This force is what adjusts the warp length.

Quantum gravity involves noncommutative geometry. In particular there are coordinates of spacetime that are not commensurate or which have an uncertainty relationship. This can be seen with the argument Bohr gave at the Solovay conference in 1930. Einstein argued that a box containing a photon could have a door opened and the mass of the box plus photon measured. This would get around the uncertainty principle. However, the uncertainty in momentum is О"p = TgО"m, for T an interval of time, g the acceleration of gravity and О"m the uncertainty in mass. The spread in distance is О"x = cО"t, and the product in О"pО"x = TgcО"mО"t = (Tg/c)О"mc^2О"t. With О"pО"x = Д§ then О"mc^2О"t = Д§ and the uncertainty in time is О"t = (Tg/c^2)О"x.

We may now consider this with the uncertainty in time and radial distance with the metric coefficients. We have then that t' = t sqrt{1 - 2GM/r'c^2}, and for О"t = t" - t' measured between two small radial distances in a Schwarzschild metric small enough then

О"t = t(sqrt{1 - 2GM/r"c^2} - sqrt{1 - 2GM/r'c^2}) ~ t(GM/c^2)(1/r" - 1/r').

For r" - r' = О"r then

О"t ~ -tGMО"r/(rc)^2,

We now consider the spread in the potential V = -GM/r, О"V = GMО"r/r^2 - GО"M/r. This leads to

О"t ~ (t/c^2)(GО"M/r + О"V)

We now let cО"t = О"x, for x a distance related to the radius. The time t is equated to GM/rc, as above. In addition the fluctuation in the radius and mass are for the Schwarzshild radius equal, so the uncertainty in the potential GО"M/r =~ О"V. This gives us

О"x ~ 2GО"M(t/rc) = 2GО"EV(t/rc^3).

A multiplication by О"t is

О"xО"t ~ 2GО"EО"t(t/rc^3) = 2GД§(t/rc^3).

The t and r are equated and this leads to

О"rО"t = 2GД§/c^4 = 8.5x10^{-75}ms.

This is a Planck unit of area divided by c. From О"pО"x = Д§ + F(x)О"xО"t it is easy to see that

О"pО"x = Д§ + (GД§/c^4)F(x)

= Д§(1 + GF(x)/c^4)

We then see that the uncertainty in the spacetime variables is due to this force. This force is what is responsible for the brane tension. If this tension is zero or minimal then the fluctuations on our brane are minimal. Measurements across huge distances are using a telescope to measure across vast distances. An extreme high energy experiment imparts huge transverse momenta/energy and increases the brane tension.

The entropy force of gravity involves motion of a holographic screen along a direction perpendicular to the screen. Here the concern is with the dynamics transverse to the motion derived by the entropy force of gravity. This means that we have a general problem with fluid dynamics. This seems to suggest that maybe the general form of fluid dynamics is given by the 3x3 matrix, maybe diagonalized by the E6.

LC