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Primitive Structures of Matter

The use of the radar method should not be taken to imply that radar is fundamental definition of distance. In relational quantum gravity, the fundamental structures of matter is a plenum consisting of arrangements of particles described by Feynman diagrams. The background in a Feynman diagram has no mathematical meaning, so that spacetime background has no physical meaning. Distances are emergent quantities arising from the structure of the plenum. Within our immediate environment, the stable structures of matter are bound by the interchange of photons. Radar provides a direct measurement of distance because it uses the same physical process, two way photon exchange, as is seen in diagrams for stable configurations of matter. Thus, we may understand the concept of distance (as defined by the radar method) as a measure of the binding between the particles constituting a stable configuration of matter.

These notional diagrams restore time ordering between the interactions, showing photon exchange as a two way process. Such a diagram is at best a rough approximation to the truth. There is no fixed time interval between absorption and emission, and we can only describe the radius of the orbit of an electron with a probability amplitude, not a numerical value. Systems containing larger numbers of particles also contain greater structure. For large structures the uncertainties in position are small compared to the scale of the structure. Spacetime structure emerges as a macroscopic property of systems containing many particles.

The Physical Content of Einstein’s Field Equation

The physical content of Einstein’s Field equation has been famously summarised by John Archibald Wheeler». The first part is straightforward: in words, Einstein’s field equation,
simply says the Einstein curvature tensor is proportional to the stress energy tensor, i.e. “matter tells space how to curve”. The second part, “space tells matter how to move”, is given by the contracted Bianchi identity
from which follows the law of local energy-momentum conservation,

Space has no meaning in Feynman diagrams and emerges in the macroscopic properties of structures described mathematically as graphs». Conservation of energy and momentum was derived from the integral formulae for probability amplitudes associated with these graphs, and follows from the principle that the interactions of elementary particles are always and everywhere the same. The view that space is an emergent property of the interactions of particles casts a deeper light on Einstein’s field equation. Instead of the duality between matter and space described by Wheeler, at a fundamental level we have only matter. Conservation of energy-momentum constrains the structure of spacetime, not the other way around.

According to general covariance a tensor equation is required in the classical correspondence to describe curvature on the emergent structure of spacetime. Since Einstein’s field equation holds for a single gravitating particle, the same tensor equation is expected to hold universally. We may also argue heuristically that each particle induces Einstein curvature equal to stress-energy, and we may therefore expect that Einstein’s equation holds generally, but there are considerable problems with formalising this argument.

The problem that, according to Einstein Field equation, changes in the matter distribution change the manifold is addressed by coordinates defined from lightspeed = c, which are independent of the mass distribution. We can embed the diagram for the Schwarzschild geometry surrounding a single gravitating particle into a Penrose diagram for the universe. In these coordinates each gravitating particle has zero size and induces Einstein curvature equal to stress energy, but this does not convey either the full content of Einstein’s field equation, nor the quantum nature of elementary particles (which are not, in general, simultaneously in eigenstates of position).

The Einstein field equation is non-linear. The curvature tensor contains products of derivatives as well as second order derivatives, and the stress-energy tensor contains products of the components of the velocities of the individual particles which contribute to it, not just on their individual masses. Consequently there is no simple way to combine solutions. If, for example, the geometries given by
and
are solutions of the field equation, we cannot conclude that
is also a solution.

The argument also neglects any contribution from the electromagnetic field or from uncharged particles. According to locality, an interaction, for example the emission of an electron by a photon, cannot be detected outside the light cone. This suggests that there is some sense in which we may say that an interaction cannot change the total scalar curvature of a region containing it. We may conjecture that, since a conservation law applies to energy momentum, it is natural that Einstein’s field equation should be preserved in all interactions.

Notwithstanding these difficulties, there remains a heuristic argument that the underlying cause of curvature is an effective small time delay in the interactions of elementary particles. It can be seen for individual particles that a small time delay causes curvature, and we may apply the argument in approximation within a sufficiently small region of each gravitating particle where changes in gravitational redshift due to the particle dominate. The precise manner in which the curvature due to each particle combines to create the properties of the manifold is non-linear and remains obscure. Analysis would require a substantial programme of research into the properties of the plenum. Such a programme should also take into account the fundamental property of spin, and would modify Feynman diagrams by the addition of label for an interval of proper time on each fermion line. Feynman diagrams would then share mathematical properties with spin networks», which are used loop quantum gravity», but unlike a simple spin network we also require photon lines labeled by zero, and we require arrowed fermion lines.

The Cosmological Constant

The cosmological constant does not appear in the calculation for curvature due to a single gravitating particle». This is not sufficient to show that the cosmological constant is zero, but it does show that there is no known mechanism by which it can be non-zero. Such a mechanism may be possible. For example, in a universe with less than critical mass for closure, one might invoke an principle» to introduce a cosmological constant to ensure a finite universe. However, such arguments generally hold little sway in physics.

An unjustified term in an equation should be considered as a “fiddle factor”, and best regarded as meaning that the theory behind the equation contains an unknown fault, rather than that the term necessarily models reality. While a non-zero cosmological constant is indicated in the analysis of observation using standard general relativity, we will see in Cosmological Implications that the teleconnection model gives at least as good a match with data while setting Λ = 0.

Non-Quantisability of Einstein’s Field Equation

It is tempting to try to define a curvature observable as a Hermitian operator with eigenvalues given by
but this is incorrect. Quantum theory is defined using plane wave motions on a Penrose diagram. The factor k is only invoked to determine the redshift between the initial and final states. k, and hence also G^ab, is a parameter, not an operator on Hilbert space. It specifies a relationship between Hilbert spaces defined at different times, and has no meaning regarding the measurement of a state at given time.

After replacing stress energy with an observable operator, we may consider the equation
This can only be true in approximation, because k and G depends on the actual, but intrinsically unknown, distribution of matter, not on the quantum state which only describes our knowledge of the distribution. Since we cannot know the exact matter distribution, we cannot know the precise values of k and G at all points of a Penrose diagram. Because k and G are parameters, not operators on Hilbert space, uncertainty in their values is modeled not by standard quantum theory. Because the actual distribution of particles is intrinsically unknown, we cannot write down an exact equation for k or G. We can calculate the probability for where a particle will be found, and in principle we can calculate probability densities for k and G. In practice the calculation is academic. Variations in the gravitational field due to quantum fluctuations are tiny compared to the resolution of gravitational measurement.

The Page-Geilker Experiment

Page and Geilker placed a massive body inside a box, with a mechanism to control its position at A or B depending on the result of a quantum process with a fifty-fifty outcome. They argued that if the classical gravitational field depends on its quantum wavefunction, its gravitational attraction should point toward some intermediate ‘average’ location. This was not observed, so they conclude that measurement of the gravitational field is equivalent to measurement of position. But this means that there is an instantaneous collapse, and hence an instantaneous change takes place in the manifold exceeding the speed of light. Page and Geilker described this as Indirect Evidence for Quantum Gravity», but the argument is not conclusive because in information theoretic interpretations, such as relational quantum gravity, the wave function describes what is known of the matter distribution, not the actual matter distribution, and the result is as expected.

The Eppley-Hannah Thought Experiment

In a famous thought experiment, Eppley and Hannah proposed that gravitational waves are used to determine the position of a particle initially in a state of poor localisation and precise momentum. They argued that, if gravitational waves behave classically then, in principle, waves of indefinitely high frequency and indefinitely low intensity can be used. If the measurement causes collapse, then position is determined to an accuracy dictated by frequency, while the change in momentum is determined by intensity and will be small. Provided that momentum is conserved, then the uncertainty principle will be violated. If, on the other hand, measurement does not cause collapse, then there exists a direct observation of the (poorly localised) wave function. In this case instantaneous collapse could be observed in principle by performing a standard measurement of position, and the speed of light would be exceeded. They concluded that the gravitational field cannot be classical without violating accepted principles of physics, and must therefore satisfy the principles of quantum mechanics.

The validity of this thought experiment has been challenged, for example by Mattingly, Phys.Rev. D73 (2006) 064025», who calculates that Eppley and Hannah’s experimental apparatus cannot be built from any feasible materials, and would be so massive as to be included within its own Schwarzschild radius. Other issues also cast doubt on Eppley & Hannah’s conclusion. Firstly, as discussed above, uncertainty in the gravitational field is not modeled by standard quantum theory, but by some extension to it. Secondly, the experiment assumes the scattering of a gravitational wave will take place by analogy with the scattering of an electromagnetic wave, but this assumption preempts the conclusion. A classical gravitational wave is a solution of Einstein’s field equation. While the field equation is non-linear and we do not have an exact solution, the assumption that scattering takes place by analogy with electromagnetic radiation does not appear justified. The detection of the position of a particle due to a perturbation in the gravitational field may be no different in principle from the detection of the position of the massive body in the experiment of Page and Geilker.

Einstein’s Field Equation ↑ Cosmological Implications →