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Structure of Spacetime

In classical mechanics, the structure of space and time is assumed as a prior property of the universe. Relational Quantum Gravity dispenses with this assumption and looks instead at the physical processes from which we find the numerical values of spacetime coordinates. The structure of spacetime is found from a study of these processes. Of course, there are numerous ways by which we may determine distance, but a particular form of measurement, the radar method, will be used as a basis for the analysis. In principle, any form of measurement could be used, but any method must lead to the same results or it would not be measuring the same physical quantity. The radar method leads to a particularly simple treatment of special relativity. On a deeper level, the mechanism it uses, the reflection of an electromagnetic signal, embodies the fundamental process described in qed», namely photon exchange, which leads to the electromagnetic force and governs all the structures in our immediate environment.

What is the Time?

There is room for confusion between two very similar questions, ‘What is time?’ and ‘What is the time?’. The first question has something to do with consciousness, and our perception of time as a flow from past to future. It admits no easy answer, but is quite distinct from the second question and only the second question is relevant in the definition of spacetime coordinates. The answer to the question ‘What is the time?’ is always something like 4:30 or 6:25. The time is a number read from a clock.

There are many different types of clock, but every clock has two common elements, a repeating process and a counter. The rest of the mechanism converts the number of repetitions to conventional units of time. The unit of time is a count of oscillations, not an absolute flow like Newtonian time». A good clock should provide accurate measurement and it should give a uniform measure of time. We cannot count less than one repetition of the process in the clock, so for good resolution the process must repeat as rapidly as possible. In a uniform clock, the repeating process must repeat each time identical to the last, uninfluenced by external matter. Clocks may be called identical if they measure identical units of time when they are together, and if moving them does not noticeably affect the physical processes of their operation. More strictly, identical clocks can be ensured using the general principle of relativity, by building them using an identical physical oscillation.

Minkowski Coordinates

A clock tells us the time, but only in its own locality. If we are to measure the time and distance of an event spacially separated from ourselves, then information must travel between us and the event. If we know the speed of information transfer, we can determine the time and distance of the event. To talk of coordinates for events we need a definition of distance, and a definition of time at a distance from the clock. Both are provided for by the radar method.

To use radar we must know the speed of light (if distance were defined using a ruler, then to measure the time at an event we would still need to know the speed of a message from the event). But now we have a paradox. To measure speed we conduct a time trial over a measured distance, but first time must be defined at both ends of the ruler, which requires knowledge of the speed of light. We know no other way to measure the time of an event at a distance from a clock; if we synchronise two clocks by bringing them together, we have no guarantee that they remain synchronised when they are separated unless light is used to test their synchronisation. To resolve the paradox we must find something fundamental, and base everything else on it. The speed of light is an absolute constant because all measurement is based upon it; coordinates remote from the origin depend for their definition on the physical behaviour of light, not the other way about.

The essential insight in relativity is that coordinates are defined from operational definitions, that is by the physical method used to determine them, not as prior properties of some abstract entity "space" which we attempt to determine through measurement. Many treatments, particularly treatments of general relativity, overlook this simple fact and replace prior "space" with "spacetime". This ignores the physical insight and reduces the subject to the manipulation of formulae and the calculation of results. In relational quantum gravity, this insight is fundamental not only to the special and general theories of relativity, but also to understanding and interpreting quantum theory.

Any method of distance determination may be used, provided that it is calibrated to (and so equivalent to) other methods. The radar method defines distance in units of time, so that the speed of light is 1 and speed is measured as a fraction of the speed of light. To restore conventional units, substitute v→v/c. An experiment to determine the speed of light actually measures the conversion factor between conventional and natural units in which the speed of light is 1.

In spherical coordinates», the position coordinates, (r, θ, φ), are determined through radial distance r, zenith angle θ from a fixed axis, and azimuth angle φ from a second axis perpendicular to the first. By convention, radial distance is the 1-direction, zenith angle is the 2-direction and azimuth angle is the 3-direction. The position coordinates are (r, θ, φ). Cartesian coordinates, (x, y, z), are found from spherical coordinates by simple trigonometry,

Maximal Speed of Information

In a more rigorous treatment, spacetime structure would not be based on the speed of light, but on a more abstract notion, the maximal speed of information transfer. In practice, this is the same as the speed of light, but experiment cannot entirely eliminate the possibility that the photon has a very small mass and that light travels slightly slower than the maximal speed. Using the maximal speed of information we would find exactly the same formulae.

We can argue that a maximal speed of information is a physical necessity. Either there is a maximal speed or there is not. If there were not, instantaneous transmission would be possible, at least in a limiting case. It would then be possible to define equal time in an absolute sense. Similarly we would have a definition of absolute space. In this case special relativity would not hold, and we would find a universe obeying different properties from those we observe. The significance to the philosophy of science» is that, perhaps for the first time in history, fundamental scientific truth can be shown by deduction (from operational definitions) rather than by induction», and scientific law can be considered proven in the strict sense.

Spacetime Diagrams

Spacetime diagrams are defined such that lines of equal time are horizontal, lines of equal distance are vertical, and light is drawn at 45°. The radar method also measures direction. Although the diagram shows only one space direction, the algebra is formally identical for 3-vectors in Cartesian coordinates; spacetime diagrams are best understood as showing the radial coordinate in 3 dimensions and do not give only a one dimensional treatment.

There is no fundamental difference between the matter in a space craft and the matter in the Earth. The space craft can be regarded as stationary, and the Earth as moving. Coordinates are set up by an observer in on the spacecraft, in exactly the same way as on Earth.


Coordinates defined in a uniformly moving spacecraft (red), as they appear to us on Earth (blue). Uniform motion means motion shown by a straight line on a spacetime diagram. The spacecraft's coordinates are also based on the behaviour of light. Lines of equal time for the spacecraft are found by drawing a line from an event to the mid-point between the spacecraft's emitted and returning signals.












Our coordinates as they appear on the spacecraft's spacetime diagram. From the viewpoint of the spacecraft, light is still drawn at 45°, and our coordinate system is distorted. The manner of distortion is exactly the same, but reversed.



A spacecraft is uniformly moving in the Earth's reference frame. The spacecraft and the Earth have identical clocks and communicate with each other by radio or light. The Earth sends the space craft two signals at an interval t. The spacecraft receives them at an interval kt on the spacecraft's clock.









Similarly if the observer on the space craft sends two signals at an interval t on his clock, they are received at an interval k't on the Earth.

k is immediately recognisable as the Doppler redshift» factor, k = 1 + z (by considering the signals as the start and stop of a burst of light of a set number of wavelengths of a set frequency). In Minkowski spacetime, redshift between inertial frames is both constant and equal for both observers, k = k'. This condition is not universally true. But we expect k = k' when the spacecraft passes close to the Earth, since otherwise there would be some fundamental difference between measurement on the spacecraft and measurement on the Earth. This is the condition for the application of the special, rather than the general, theory of relativity.

Time Dilation

The space craft and the Earth set both clocks to zero at the moment the space craft passes the Earth. The space craft is moving at speed v, so by definition, after time t on the Earth clock, the space craft has travelled distance vt. So, Earth’s signal was sent at time t – vt, and returned at time t + vt. Using Doppler shift, if the Earth sends signals at an interval t – vt the space craft receives them at an interval k(t – vt). So, By applying the Doppler shift for the signal coming back: Rearranging: Substituting k into the formula for T, we find the time measured by a spacecraft’s clock during an interval t on the Earth’s clock is given by:

Lorentz Contraction

The bow and stern of the spacecraft are shown as parallel lines. The space craft's clock is in the bow. The space craft and Earth set their clocks to zero when the bow passes the Earth clock. Earth uses radar to measure the distance, l, from bow to stern, by sending a signal at time –l, which returns at time l on the Earth clock. The same signal is used to determine the proper length, L, as measured on the spaceship. By the Doppler shift, the outgoing signal passes the bow at time –l / k on the space craft's clock, and the returning signal reaches the bow at time kl. So, as measured on the spacecraft: Substituting in the formula for k and rearranging gives the Lorentz contraction:

4-Vectors

A spacetime vector, or 4-vector, can be described as the difference in the coordinates of two events (like a displacement vector in 3 dimensions). Conventionally a spacetime vector is written
where x⁰ = t is the difference in time coordinates of the events, known as the time component, and (x¹, x², x³) is an ordinary vector in 3 dimensions, or a 3-vector. A 4-vector can be used to represent the motion of an object on a spacetime diagram.

There is a particular coordinate system in which the body is at rest. The velocity 4-vector is (1, 0, 0 0) in these coordinates. Using the formula for time dilation, we can find the time coordinate of this vector in a coordinate system moving at 3-velocity, v = (v¹,v²,v³). The space coordinate is found by multiplying 3-velocity by time. Then the velocity 4-vector of an object moving with 3-velocity v is:


Lorentz transformation» is performed generally by matrix multiplication (see also the derivation by Einstein»).


Using the Lorentz factor, 4-velocity simplifies to (γ, γv).

E = mc²

Since momentum, (p¹, p², p³) is conserved in Newtonian mechanics, it is a vector with central importance. It is natural to define a corresponding 4-vector p = (p⁰, p¹, p², p³). Momentum is mass times velocity. So, 4-momentum is p = (γm, γmv), and the time component of 4-momentum is:
Convert this to conventional units, by substituting v→v/c :
is the classical expression for kinetic energy, so this is an expression for energy. Einstein removed the arbitrary constant in classical energy by putting E=p⁰c². We find, for the energy of a body at rest,
thereby establishing the equivalence of rest mass and energy.

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