KJW

Here are the masses (MeV/c²) of the three charged leptons: electron: 0.51100 muon: 105.658 tauon: 1776.84 These are three closely related fundamental particles. If there is any pattern among the masses of particles, it should here among the charged leptons. [Source: https://en.wikipedia.org/wiki/Lepton#Table_of_leptons]

This assumes that you know, at least approximately, the value of π.

Perhaps you can elaborate on what an affine parameter is, particularly with regards to null geodesics, which cannot be parametrised with proper time or arc length. An affine parameter is a parameter along a geodesic that preserves the geodesic equation's form. For null geodesics, proper time τ is zero, so we can't use it. Instead, we use an affine parameter λ, which labels points along the path in a way that keeps the motion equation: [math]\dfrac{d^2 x^\mu}{dλ^2} + \Gamma^\mu_{\nu\sigma} \dfrac{dx^\nu}{dλ} \dfrac{dx^\sigma}{dλ} = 0[/math] This ensures the particle’s path remains a true geodesic, even without proper time. In what way is this not circular? That is, you provide the null geodesic equation in which the parameter is an affine parameter, but when asked what an affine parameter is, you refer back to the null geodesic equation.

I think it is worth saying that every situation in which π appears is in some way connected to a circle. To see that this is true, for any given expression in which π appears, consider why it is π that appears in the expression. This means tracing the number back to its definition, which was originally based on the circle. For example, consider the gamma function of a half-integer: [math]\displaystyle \Gamma(\dfrac{3}{2}) = \int_{0}^{\infty} \sqrt{t}\ \exp(-t)\ dt[/math] [math]\displaystyle \text{Let } t = x^2\ \ \ \ ;\ \ \ \ dt = 2x\ dx[/math] [math]\displaystyle \Gamma(\dfrac{3}{2}) = \int_{0}^{\infty} 2x^2\ \exp(-x^2)\ dr[/math] [math]\displaystyle = \int_{0}^{\infty} \exp(-x^2)\ dx\ \ \ \ \ \text{(integration by parts:}\ u=x,dv=2x\exp(-x^2)\ \text{)}[/math] [math]\displaystyle \int_{0}^{\infty}\int_{0}^{\pi/2} r\ \exp(-r^2)\ d\theta\ dr = \dfrac{\pi}{4} \int_{0}^{\infty} 2r\ \exp(-r^2)\ dr[/math] [math]\displaystyle = \dfrac{\pi}{4} \int_{0}^{\infty} \exp(-t)\ dt = \dfrac{\pi}{4}[/math] [math]\displaystyle \int_{0}^{\infty}\int_{0}^{\pi/2} r\ \exp(-r^2)\ d\theta\ dr = \int_{0}^{\infty}\int_{0}^{\infty} \exp(-x^2-y^2)\ dx\ dy[/math] [math]\displaystyle = \int_{0}^{\infty} \exp(-x^2)\ dx\ \cdot \int_{0}^{\infty} \exp(-y^2)\ dy = \dfrac{\pi}{4}[/math] [math]\displaystyle \text{Therefore }\ \ \Gamma(\dfrac{3}{2}) = \int_{0}^{\infty} \exp(-x^2)\ dx = \dfrac{\sqrt{\pi}}{2}[/math] [If the above LaTex doesn't render, please refresh browser] In this case, the connection to a circle is the use of polar coordinates to evaluate the definite integral of the Gaussian function.

Before you become too engrossed by the more speculative aspects of your document, perhaps you could reply to the following which I posted earlier: Perhaps you can elaborate on what an affine parameter is, particularly with regards to null geodesics, which cannot be parametrised with proper time or arc length.

Have you seen A Clockwork Orange?

Reading through the opening post, I thought it was going to be about the notion that the laser point, if far enough away, would be travelling faster than the speed of light.

I think it is interesting that there are two distinct, even somewhat contrary, notions of degrees of irrationality. For example, one can consider the following sequence as a sequence of increasing irrationality: rational numbers solutions of quadratic equations with integer coefficients solutions of cubic equations with integer coefficients solutions of quartic equations with integer coefficients ... solutions of polynomial equations of finite degree n with integer coefficients ... transcendental numbers Or alternatively, one can consider how well a given number can be approximated by a rational number relative to the size of the denominator of the rational number, as indicated by the irrationality exponent [math]\mu(x)[/math] in the expression: [math]\left|x - \dfrac{p}{q}\right| < \dfrac{1}{q^{\mu}}[/math] Quite remarkably, the Liouville numbers, which are transcendental numbers, have infinite irrationality exponent and are most closely approximated by rational numbers, whereas the golden ratio, a solution of a quadratic equation, with irrationality exponent 2, is poorly approximated by rational numbers, being an extreme case of the Hurwitz inequality: [math]\left|x - \dfrac{p}{q}\right| < \dfrac{1}{\sqrt{5}q^2}[/math] Even worse than the golden ratio are other rational numbers (approximated by rational numbers that are not equal to the number), which have irrationality exponent 1.

The golden ratio is not transcendental: [math]\varphi =\dfrac{1+\sqrt{5}}{2}[/math] Or putting it another way, it is a solution of the quadratic equation: [math]\varphi^{2} - \varphi - 1 = 0[/math] [If the above LaTex doesn't render, please refresh browser] The golden ratio has an interesting property: it is representative of a family of numbers that are the least rational of all the numbers. By contrast, the Liouville numbers are numbers that are more closely approximated by rational numbers than all other numbers, and they are transcendental.

All of the mathematical expression, including the opening and closing tags, needs to be on the one line. It can wrap to multiple lines, but no new line characters.

I assumed that this was simply to eliminate boundary effects that result from trying to pack circles into a finite area. This seems unnecessary to me as one can tesselate an infinite plane with hexagons and inscribe a circle into each hexagon. Note also that the hexagon is the polygon with the largest number of sides that can tesselate a plane. And given that the ratio of the area of an inscribed circle to the area of the inscribing polygon increases with the number of sides of the polygon, the hexagon is the polygon producing the maximum packing density of circles.

The packing of spheres is different to the packing of circles in that one can't arrange spheres around a central sphere in a gap-free way, whereas one can arrange six circles around a central circle and this arrangement is gap-free. And because a flat plane can be tessellated with hexagons, it is clear that the maximum packing density for circles in a plane is the ratio of the area of an inscribed circle to the area of the inscribing hexagon: Area of inscribed circle = πr2 ≈ 3.1416 r2 Area of inscribing hexagon = 6r2 tan π/6 ≈ 3.4641 r2 Ratio of inscribed circle to inscribing hexagon = [math]\dfrac{\pi}{6}\cot\dfrac{\pi}{6} = \dfrac{\sqrt{3}\pi}{6} \approx 0.9069[/math] The maximum packing density for spheres is [math]\dfrac{\pi}{3\sqrt{2}} \approx 0.74048[/math] for a number of different arrangements, all of which has each sphere touching 12 neighbouring spheres. According to Wikipedia: "In 1611, Johannes Kepler conjectured that this is the maximum possible density amongst both regular and irregular arrangements—this became known as the Kepler conjecture. Carl Friedrich Gauss proved in 1831 that these packings have the highest density amongst all possible lattice packings. In 1998, Thomas Callister Hales, following the approach suggested by László Fejes Tóth in 1953, announced a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving checking of many individual cases using complex computer calculations. Referees said that they were "99% certain" of the correctness of Hales' proof. On 10 August 2014, Hales announced the completion of a formal proof using automated proof checking, removing any doubt." [If the above LaTex doesn't render, please refresh browser.]

Perhaps you can elaborate on what an affine parameter is, particularly with regards to null geodesics, which cannot be parametrised with proper time or arc length.

From https://en.wikipedia.org/wiki/Greisen%E2%80%93Zatsepin%E2%80%93Kuzmin_limit "The Greisen–Zatsepin–Kuzmin limit (GZK limit or GZK cutoff) is a theoretical upper limit on the energy of cosmic ray protons traveling from other galaxies through the intergalactic medium to our galaxy. The limit is 5×1019 eV (50 EeV), or about 8 joules (the energy of a proton travelling at ≈ 99.99999999999999999998% the speed of light). The limit is set by the slowing effect of interactions of the protons with the microwave background radiation over long distances (≈ 160 million light-years)."

Don't forget that you still haven't answered my question yet.

On page 37, you said that a test particle of mass m follows: [math]\dfrac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\nu\sigma} \dfrac{dx^\nu}{d\tau} \dfrac{dx^\sigma}{d\tau} = 0[/math] However, this equation does not work for a test particle of mass zero (eg a photon). How would you adjust this equation so that it does work for a test particle of mass zero? [If the above LaTex doesn't render, please refresh browser.]

No, I'm stating that while the loopholes of local hidden variables were closed by the experimental work by John Clauser, Alain Aspect, and Anton Zeilinger, who won the Nobel prize for their work, the assumption for the closure of those loopholes is a 3+1D space-time. I demonstrate that with Feynman integrals, a 3+1+1D geometry could be hidden because that path is usually canceled out. Even the no-communication theorem assumes a 3+1D space-time. Extending quantum dynamics to an extra dimension through a quantum behavior isn't reaching, but examining, or exploring, hypothetical perspectives not approached before. I need some clarification: Are you claiming that in the extra dimension, entangled particles are local where they are measured even though they may be very distant in 3+1D spacetime? If so, then you are assuming that there is a need for the communication of measured results between the particles. Where in the proof of the no-communication theorem does it indicate the assumption of 3+1D spacetime?

One thing worth mentioning: A single photon travelling in free space cannot gravitationally collapse, regardless of how much energy it has. Any photon exists in frames of reference where is has any particular value of energy. Since this energy can be arbitrarily high, there can be no gravitational collapse. Also, a single photon travelling in free space cannot split into other particles because this would violate the conservation of energy-momentum. However, a single photon can collide with some object, or two photons travelling in opposite directions can collide with each other, to produce... whatever.

For two particles to be quantum entangled, two conditions need to be satisfied: (1): The states of the two particles are correlated. That is, the state of one particle depends on the state of the other particle. (2): The combined state of the two particles is a quantum superposition. If only (1) is satisfied, the two correlated particles may be classical particles (eg billiard balls). Or it may be the result of measuring an entangled pair of particles, causing the quantum superposition to "collapse" (Copenhagen interpretation). If only (2) is satisfied, the particles are individually a quantum superposition, and therefore the combined state of the two particles is also a quantum superposition, but the two particles are completely independent of each other, perhaps because they are in different galaxies. It is worth noting that an arbitrary two-particle state is most likely to be an entangled state. However, the entangled states usually encountered in entanglement experiments are special entangled states, not the run-of-the-mill arbitrarily chosen entangled two-particle states. (In mathematical terms, the Hilbert space of the two-particle states has a higher dimensionality than the Hilbert space of the corresponding non-entangled two-particle states.)

That's not quite true. The CMBR represents the time the universe became transparent to electromagnetic radiation. However, it is proposed that before then, there was a time when the universe became transparent to neutrinos, and therefore the cosmic neutrino background may be something before the CMBR that could in principle be studied.

We are in agreement about the need of quantum mechanics to have an ontology. The "shut up and calculate" viewpoint is wholly unsatisfactory. However, we do disagree about the ontology itself, and maybe also about the relationship between mathematics and physics. The ontology I see is something like the many-worlds interpretation. I say "something like" because I haven't quite worked out the precise details, and I feel that the usual presentation of the many-worlds interpretation is problematic, although it is not clear to me if these problems extend to the original relative state formulation by Hugh Everett. Nevertheless, I do regard the wavefunction to be not so much about a particle, but about the entire reality in which the particle exists, including the entire history and future of the reality. Then the reality that includes the [math]|\psi_1\!\!>[/math] state also includes the [math]|\phi_1\!\!>[/math] state, and the reality that includes the [math]|\psi_2\!\!>[/math] state also includes the [math]|\phi_2\!\!>[/math] state. Thus, non-locality is taken care of without invoking any interaction between the two states. Note that the no-communication theorem says that there is nothing that can be done to the [math]|\psi\!\!>[/math] state that will affect the result of a measurement of the [math]|\phi\!\!>[/math] state. That is, although the correlation between distant states is strange, it is not strange enough to require interaction between the distant states. I believe that the ontology of quantum mechanics needs to connect with the mathematics of quantum mechanics. The notion that there is an underlying physical mechanism that is not a part of the mathematics seems wrong to me. I didn't discuss the violation of Bell's inequalities because I was discussing the nature of quantum entanglement. I wasn't trying to prove that quantum physics cannot be explained classically. Bell's inequalities aren't exactly about quantum entanglement. They are limitations on the properties of classical states. Their violation in quantum entanglement experiments demonstrate that the correlations that occur between quantum entangled states cannot be explained in classical terms. But this is more about the nature of quantum states than about entanglement. There is no mention of locality within Bell's inequalities. The notion of locality is merely to close a loophole in Bell's theorem concerning the possibility of communication between entangled distant states. Thus, it seems to me that by making the distant states local via an extra dimension, you are claiming that it is necessary for the result of a measurement of one state to be communicated to the other state. The mathematics does not seem to point to this notion. The correlation between quantum entangled states is already a part of the multi-particle state and doesn't need to be reinforced by a communication of measured results between states. [If the above LaTex doesn't render, refresh the page]

MythBusters did an episode in which something like this was tested: MythBusters Episode 217: Household Disasters Premier Date: July 17, 2014 ... A dog bowl can focus the sun’s rays at a small enough point to start a fire due to it being very hot.CONFIRMED The Build Team sets up a table on a wooden deck outside on a sunny day, made to look like a picnic, and put out highly flammable objects to improve their chances. This includes dried flowers, decaying wood, and paper. They set two types of dog bowls, metal and glass, in various sizes to focus the sun’s rays. Due to high humidity and wind, they use a theatrical light to warm up the set, replicating ideal conditions of the summer. The temperature of the set was 105 °F (41 °C), with 12% humidity. After two minutes, the glass bowls started a fire on the set, confirming the myth under ideal conditions. (This segment was not shown in the U.S. broadcast.) ... There was also the 20 Fenchurch Street skyscraper in London, nicknamed "The Walkie-Talkie", which during its construction, due to its concave reflective surfaces, focused sunlight onto the street below, causing damage to parked vehicles.

"Residual Order" Equation for Tensors I couldn't think of a better name for this topic. Anyway, it is about a concept in tensor calculus similar to dimensional analysis in physics. In physics, one has the dimensions of length, time, mass, etc. In geometry, there is only length. However, there are other "weights", as well as "orders". The "residual order" equation expresses a relation between these, and indicates what constitutes a <b>properly definable quantity</b>. First, it is necessary to define some terms (most of which I coined):<br><br> For the tensor T^{i1 ... ip} _{j1 ... jq} (and other indexed quantities properly expressed as such):<br><br> Define the CONTRAVARIANT ORDER as p.<br> Define the COVARIANT ORDER as q.<br> Define the TOTAL ORDER as p+q.<br> Define the RESIDUAL ORDER as p-q.<br><br> It is the RESIDUAL ORDER that is of interest here. The residual order has an important property: it is unchanged by the contraction of any number of pairs of indices (because q is decreased by the same amount as p).<br><br><br> Every tensor possesses three weights:<br><br> (1): TENSOR WEIGHT<br><br> For tensor <b>A</b>, transforming as <b>A</b>' = <b>A</b> |<font face="Symbol">¶</font>x^j/<font face="Symbol">¶</font>x'^s|^T ...(etc) under a coordinate transformation, define the TENSOR WEIGHT as T.<br><br> (2): DIMENSION WEIGHT<br><br> For tensor <b>A</b>, transforming as <b>A</b>' = k^D <b>A</b> under the change of length units dx'ⁱ = k dxⁱ (not a coordinate transformation), where k is an arbitrary non-zero constant, define the DIMENSION WEIGHT as D. This corresponds to the length dimension in physics. However, in geometry, this is the only dimension and all other dimensions of physics can be expressed in terms of length by the application of the fundamental constants.<br><br> (3): METRIC WEIGHT<br><br> For tensor <b>A</b>, transforming as <b>A</b>' = k^M <b>A</b> under the change of scale g'^ij = k g^ij, where k is an arbitrary non-zero constant, define the METRIC WEIGHT as M. Although this is called the "metric" weight, note that the scale transformation is applied to the INVERSE of the metric tensor. This is to maintain a convention that seems to exist throughout tensor calculus.<br><br> These weights can also be defined for quantities that are not tensors.<br><br><br> STANDARD WEIGHTS are quantities that have a weight of +1 or -1 for their particular weight type, but have a weight of 0 for the other weight types.<br><br> TENSOR: The permutation symbols e^{i1 ... in} and e_{i1 ... in} (n = dimension of the space) have a TENSOR weight of +1 and -1, respectively. The DIMENSION and METRIC weights are both 0.<br><br> DIMENSION: The differential dxⁱ and the partial differentiation operator <font face="Symbol">¶/¶</font>xⁱ have a DIMENSION weight of +1 and -1, respectively. The TENSOR and METRIC weights are both 0.<br><br> METRIC: The inverse of the metric tensor g^ij and the metric tensor g_ij have a METRIC weight of +1 and -1, respectively. The TENSOR and DIMENSION weights are both 0.<br><br><br> The four orders and three weights defined above separately obey two rules:<br><br> (1): Only quantities of the same weight (or order) can be added, subtracted or equated. There exists a zero quantity for all orders and weights.<br><br> (2): For the weights (or orders) W(A) and W(B) of quantities A and B, the weight of the outer product AB, W(AB) = W(A) + W(B). However, the RESIDUAL ORDER and the TENSOR, DIMENSION and METRIC weights are unchanged by the contraction of any number of pairs of indices. Therefore, the outer product can be combined with the contraction of any number of indices without changing the value of the RESIDUAL ORDER or the TENSOR, DIMENSION and METRIC weights.<br><br><br> The Residual Order Equation:<br><br> For RESIDUAL ORDER = R, TENSOR WEIGHT = T, DIMENSION WEIGHT = D, and METRIC WEIGHT = M:<br><br> R = nT + D + 2M <br><br> (n = dimension of the space)<br><br><br> Examples:<br><br><font face="Courier"> . R . T . D . M . Quantity<br> --------------------------<br> . n . 1 . 0 . 0 . Permutation symbol e^{i1 ... in}<br> .-n .-1 . 0 . 0 . Permutation symbol e_{i1 ... in}<br> . 1 . 0 . 1 . 0 . Differential dxⁱ<br> .-1 . 0 .-1 . 0 . Partial differentiation operator <font face="Symbol">¶/¶</font>xⁱ<br> . 2 . 0 . 0 . 1 . Inverse of the metric tensor g^ij<br> .-2 . 0 . 0 .-1 . Metric tensor g_ij<br> . 0 . 0 . 0 . 0 . Kronecker Delta (identity matrix) <font face="Symbol">d</font>ⁱ_j<br> . 0 . 0 . 0 . 0 . Generalised Kronecker Delta <font face="Symbol">d</font>ⁱ¹_j1^._.^._.^._.^ip_jp<br> . 0 . 2 . 0 .-n . Determinant of metric tensor g<br> . 0 . 0 . 1 .-½ . Arc-length s<br> . 1 . 0 . 0 . ½ . Unit tangent vector (n-velocity) dxⁱ/ds<br> . 0 .-1 . n . 0 . Volume element dV<br> . 0 . 0 . n .-½n. Volume <font face="Symbol">ò</font>_R |g|^½ dV<br> .-3 . 0 .-1 .-1 . Partial derivative of metric tensor <font face="Symbol">¶</font>_ig_jk<br> .-1 . 0 .-1 . 0 . Christoffel symbol <font face="Symbol">G</font>_ij^k<br> .-2 . 0 .-2 . 0 . Riemann tensor R_ijk^l<br> .-4 . 0 .-2 .-1 . Riemann tensor R_ijkl<br> . 0 . 0 .-2 . 1 . Riemann tensor R_ij^kl<br> . 0 . 0 .-2 . 1 . Magnitude of Riemann tensor |g^ir g^js g^kt g_lu R_ijk^l R_rst^u|^½<br> .-2 . 0 .-2 . 0 . Ricci tensor R_ij<br> . 0 . 0 .-2 . 1 . Ricci tensor R_i^j<br> . 2 . 0 .-2 . 2 . Ricci tensor R^ij<br> . 0 . 0 .-2 . 1 . Ricci scalar R<br> .-2 . 0 .-2 . 0 . Einstein tensor G_ij<br> . 0 . 0 .-2 . 1 . Einstein tensor G_i^j<br> . 2 . 0 .-2 . 2 . Einstein tensor G^ij<br> .-2 . 0 .-2 . 0 . Weyl conformal tensor C_ijk^l<br> .-4 . 0 .-2 .-1 . Weyl conformal tensor C_ijkl<br> . 0 . 0 .-2 . 1 . Weyl conformal tensor C_ij^kl<br> .-3 . 0 .-3 . 0 . Covariant divergence of Weyl conformal tensor <font face="Symbol">Ñ</font>_uC_ijk^u<br> . 0 . 0 . 0 . 0 . The topological invariant <font face="Symbol">ò</font>_R R_{[i1 i2} ^{i1 i2} ... R_{in-1 in]}^{in-1 in} |g|^½ dV</font><br><br><br>

Maybe that's what's happening with Trump.