Logo: Proper Interval Locality
Unified variational theory explores what happens if mechanics and optics are considered to be the properties of a single constituent of nature. The two branches being combined as the science of the motion of matter and its interactions; within the context of relativistic space-time. In combining the two branches of physics the new interpretation, retains the full scope of classical physics and is also effective in predicting quantum behaviour for example observations indicating the wave-particle duality of matter, the interference of light and how Bell's inequality can be violated in a universe characterised by relativity.
Reference Frames
The laws of motion of observable material bodies and light are formulated relative to reference frames. Our experience tells us that in order to specify the position of a body in space we need three coordinates. Simplest way is to define the coordinates relative to a three dimensional Cartesian rectilinear reference frame. Measurements of distance (using a rule) made relative to a given datum point in space can be represented on our coordinate reference frame. To represent events we add to a body's position space a moment in time. To be able to measure time we must also use clocks.
A point in space can be represented by (x, y, z) and an event by (x, y, z, t)
Inertial Reference Frame
In Newtonian mechanics, an inertial reference frame, is one in which Newton's first law of motion holds. The principle of special relativity takes a broader view where the inertial frame includes all physical laws, not simply Newton's first law.
In the context of unified variational theory it is important to recognise that our ability to fix positions in space and events in space and time is restricted to observable bodies. Similarly we can only be confident in the principles of relativity in as much as the apply to observable bodies. What we can infer about the hidden world of quantum objects is that our world of space, time and measurable locality is a product of their existence and what is observable is not in itself fundamental. We must remember that in order to determine the locality of observable bodies in space and time we need clocks and rules. The constancy of the locality and calibration of these devices depends on the group behaviour of the quantum entities that form them.
Our reference frames are therefore also dependent on the group behaviuor of quantum objects, but we are forced into using our inertial coordinate reference systems as the only means of formulating our models for the motion of matter and light.
It is the object of unified variational theory to establish how the hidden locality of quantum entities is projected onto our inertial reference frames. So that we can predict how quantum entities will interact with the observable world to give measurable results,in a self-consistent way
The metric of space-time
Minkowski showed that the principles of special relativity demanded that the metric characterising space time is in rectilinear coordinates: -
Image: minkowski metric
s is called the proper interval between two events that are separated by the coordinate differences delta x,delta y,delta z and delta t. The proper interval is invariant, that is its value is the same for the all inertial reference frames.
It is important to recognise that space-time characterised by Minkowski's metric is not Euclidean, note its metric signature: -
(+, +, +, -)
compared with : -
(+, +, +, +)
for a Euclidean four dimensional manifold.
This distinction is important because of the possible ways it can influence our interpretation of why the fundamental principles of variation in physics are as they are!
Note also that the metric signature (-, -, -, +) is equally valid for Minkowski Space-time. This is because although our actual measures of distance and time are always real, in the metric we find if distance is real then time is imaginary and if time is real then distance is imaginary.
This particular feature of the metric signature allows the proper interval to have zero magnitude without all the values of the coordinate differences having to equal zero.
It is this characteristic of the flat space-time metric that permits the unification of optics and mechanics.
Although shall confine ourselves to studying variational theory in flat space time, we should also observe that Schartzchild's metric for the gravitational field also has the metric signature: -
(+, +, +, -)
The actual metric in spherical coordinates being: -
Image: schwartzchild's metric
Where
a =2GM/c2
G is the gravitational constant, M is the mass of the body and c is the speed of light.
As with the Minkowski metric the Schwartzchild metric will also support the unification of the fundamental principles of variation in physics.
Principles of Variation in Physics
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