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The k-Calculus

Special relativity was described by Einstein as a principle theory, as distinct from a fundamental theory, because it derives its results from principles of measurement, and does not explain fundamental physical mechanisms. It enables us to derive the mathematical properties of Minkowski spacetime, but does not explain the underlying structure, or give the reason why light behaves as it does. In Foundations Of Special Relativity, the structure of Minkowski spacetime was found from the radar method, using two way transmission of light. As used in the radar method, light is the sensor for the properties of spacetime, not the cause of them. The radar method assumes a greater importance in relational quantum gravity. In qed, two-way transmission of photons is responsible for the electromagnetic force, and hence for the all the structures of matter we observe in our immediate environment. The radar method can use light as a sensor for the properties of spacetime because it uses the same physical process, two-way photon exchange, which is also the cause of the structure of spacetime.

The k-calculus^» was introduced by Hermann Bondi as a simple means of introducing special relativity. In the k-calculus, the metric is determined by the reflection of light (the radar method). Any other method of measurement may be used provided that it is calibrated to, and hence equivalent to, radar. In special relativity k may be identified with Doppler shift, k = 1 + z . The treatment of stationary observers generalises the k-calculus to the case in which k may be identified with gravitational redshift. This approach is directly based on the equivalence principle, has advantages in (relative) mathematical and conceptual simplicity, and leads to a formulation of general relativity which is mathematically equivalent to other formulations.

Here I generalise the radar method to allow that the minimum time for the return of a photon may depend not only on the speed of light, c, but also on an effective minimum proper time interval between the absorption and re-emission of a reflected photon. In the idealised case of a single elementary particle, neglecting spin, the geometrical effect is calculated. I will show that an effective minimal time interval of reflection proportional to the mass of the reflecting elementary particle leads the replacement of Minkowski geometry by Schwarzschild.

A Modification to Radar

Consistent with Einstein’s 1905 paper and the internationally agreed empirical definition of the metre, Bondi's k-calculus for special relativity postulates instantaneous reflection of radar at the event whose position is to be determined. Although reflection clearly takes place on a very small timescale, there is no empirical basis on which we can say it is actually instantaneous. A natural generalisation is to hypothesise a small time delay between absorption and emission in proper time of a fundamental charged particle (electron or quark) reflecting electromagnetic radiation.

An intrinsic delay between the interactions of elementary particles affects the empirical definition of spacetime measurement (e.g. SI units). We seek to analyse the geometric implications. The metric is determined as in the k-calculus for special relativity, from the minimum time for the return of information reflected at an event. But now this minimum net time depends not only on the maximum theoretical speed of information, c, but also on an effective least proper time between absorption and emission in the reflection of a photon. Let the effective time delay be 4GM, where M is the mass of the reflecting particle and 4G is a constant of proportionality. It will be seen that G may be identified with the gravitational constant. Special relativity can be recovered in the limit in which G goes to zero (allowing G to go to zero introduces the Landau pole, so this limit may not be strictly valid).

There are several reasons for introducing such a delay. Firstly, as shown here, the delay perturbs the metric, resulting in a curved spacetime obeying Einstein’s field equation. If the reflection of a photon were instantaneous, the physical metric would be Minkowski. The intrinsic time delay in reflection, 4GM, causes a small amount of curvature, in accordance with Einstein's field equation. Secondly, it is well known that some small scale modification is needed to qed in order to remove the ultraviolet divergence and resolve the Landau pole. The delay introduced here is an effective cut-off and allows the construction of a consistent qed. Finally, a minimum time between interactions proportional to mass may be related to the concept of inertia; if the interactions of muons and electrons with photons are discrete and identical, then it is natural that the acceleration due to the electromagnetic field will be proportional to the frequency of interaction. An intrinsic delay between interactions proportional to mass will result in accelerations inversely proportional to mass.

The calculation performed here applies to fundamental stable particles which can emit and absorb photons. In the real world these are charged particles with spin, electrons or quarks. It does not directly apply to macroscopic bodies. If radar is used to measure the position of the moon, for example, then an individual reflected photon can be said only to measure the position of a single electron in the surface of the moon. A classical radar pulse contains many such photons, the time delay at each reflection being dependant on the mass of an electron, not the mass of the moon. A more realistic calculation should also take into account charge and spin, and would be expected to yield Kerr-Newman geometry. It is not known how to carry out such a calculation within the k-calculus, but it appears reasonable to separate off the contributions due to charge and spin, and to regard the calculation below of a Schwarzschild geometry surrounding a particle in a position eigenstate as a genuine indicator of an effective time interval between the interactions of an elementary particles.

A Particle in a Position Eigenstate

Since I am discussing measurement of position, I will describe eigenstates of position. There is no such thing as a perfect eigenstate of position of a massive point particle. Nonetheless such states span Hilbert space and will be sufficient for a description of geometry. For the purpose of analysis, I will consider a static system and calculate the metric at particular time. I consider only a system with a single gravitating particle, at O. I will assume distance and time scales such that cosmological expansion is negligible.

An isolated elementary particle in an eigenstate of position has spherical symmetry and spacetime diagrams may be used to show a radial coordinate in n dimensions without loss of generality. A spacetime diagram is drawn, showing a tangent space with an origin at O, so that light is shown at 45°, lines of equal time are horizontal and time is proper time for the gravitating particle. In tangent space at O, the coordinate distance between Beth and the particle is r.

As shown in tangent space at O, Beth uses the radar method to determine a distance coordinate for the particle. Beth cannot resolve the points A and E where a photon is absorbed and emitted and places the particle at apparent position P. If the effective delay in the reflection is 4GM, then the coordinate coordinate distance of P from Beth is ρ = r + 2GM.

As calculated for stationary observers, the physical metric (with angular directions suppressed) is So, Beth’s clock runs fast by a factor k compared to proper time, t, for the particle at O. Minkowski coordinates (t, r) for tangent space at O are stretched by a factor k−1 in the time direction and by k in the radial direction compared to Minkowski coordinates, (T, R) for tangent space at B, as determined by Beth using the radar method. Since the radar method determines that R = T, we have So, Using the apparent position ρ as the radial coordinate, and substituting ρ, we find the Schwarzschild metric:

It will be observed that the event horizon, ρ = 2GM, maps to r = 0. Only space outside the event horizon is mapped in these coordinates. The space inside the event horizon, ρ < 2GM, has no physical meaning. A particle which is point-like in a tangent space at the position of the particle, is mapped to the event horizon in a tangent space at the position of a distant observer. The effect of the discrete interval of proper time between interactions is a singularity at ρ = 0 which “magnifies” a point-like particle to the size of the event horizon.

The Path of Light

With r as the radial coordinate, the metric is Superficially, it appears that the speed of light is less than unity in these coordinates. Light should then be drawn at a greater than 45° to the horizontal. The inconsistency is resolved because this refers to the mean velocity of light, as determined from measurement using the radar method. The true path of light is plotted at 45°. The coordinate singularity, or event horizon in the Schwarzschild geometry, is actually at the position of the particle itself, while the singularity at ρ = 0 is an illusory point, which does not correspond to a position in space.

Einstein’s Field Equation

For a single particle in an eigenstate of position, stress energy is given by
By considering the coordinate transformation r → r + δr, we replace the delta function with a uniform density over a small region. We now take the limit δr → 0 to see that Einstein’s field equation holds for the single particle case
The derivation here applies only to the case where the gravitating source is a single particle with known position. The next section considers the issues in the general case.

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