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# Coordinate systems and Relativity

I probably have a unique perspective on relativity.
I learned about it in college then spent a career working with geometries and coordinate systems, as well as physics defined by Newton and Maxwell for both motors and high speed, high precision motion control. After retiring I re-learned relativity.

In a matter of speaking a large machine tool is a 'universe' with its geometries and coordinate systems.
The foundation of the system is called the machine coordinate system.

Each moving component must have a defined range of motion or mechanically something could crash into another component or the mechanical limit for the component's motion. For example a ball screw is often used for linear motion and it cannot move beyond its limits. Rotary axes are often without a limit because after completing a rotation its position rolls over, like +359 to 0 to +1. Some rotary axes need a limit because of cabling lengths or interference so a range in degrees could be like -270 to +45.
When a machine tool is powered up, the CNC software must determine the current location for all movable components.

Each component needs a zero reference point so the basic geometry of the machine is clearly defined. Linear axes are perpendicular to others or parallel. A rotary axis is defined to rotate about a linear axis. Attachments can be mounted or removed from the machine so the collection of axes can change in time.
For a linear axis zero reference point will define at the moment of contact the current position of the component is certain within the limits of the machine.

For example the X-axis or horizontal axis could have positions from -1 mm to many + meters. The Y or vertical axis could have positions from -1mm to many + meters. The Z-axis or  depth could have positions from -1mm to + several meters.

This mechanical system with its geometry and axes is the foundation for thesoftware system. The machine tool software presents to both the programmer or the operator a part coordinate system.

This part coordinate system defines the translation of the programmer's axes (usually the letters XYZABCUVW, for the 3 primary linear motion, the 3 rotary motions, and 3 secondary linear motions).

The simplest part coordinate sytems: for a mill X0Y0 is at the lower left and Z0 is the top of the part (so Z-1.0 mm moves the drill tip 1mm into the part).
For a lathe X0 is the center of the part turning in the spindle while Z-1.0mm moves the cutting tool 1mm into the spinning part.

A fixture coordinate system can be defined for versatility. With one fixture X0Y0Z0 is one physical location in the machine coordinates  but with another fixture (invoked by the programmer or operator) the part coordate X0Y0Z0 is in a different physical location.
Software features allow the part coordinate system going through transformations such as when the rotary axis moves with a fixture on it. The programmer deals with what appears to be a static coordinate system.

In summation:

The operator or programmer is using a geometry (a part coordinate system) that is undergoing transformations in relation to the physical machine.The software can do this because all the relationships, between the part coordinate and the machine coordinates are clearly defined.
In this complex system everything is firmly rooted in the underlying machine coordinate system. However the programmer is commanding motions they are predictable and verifiable on the real machine.

The comparison to relativity:

The first big difference is the universe has no physical coordinate system. There is no fixed zero reference  point in the universe to establish a geometry with defined planes and dimensions. Every geometry must be based on the observer; the observer picks a geometry with its planes and the observer picks the zero reference point for all dimensions. For example, the initial time dimension value could be zero, so all time measurements are since the observation began, or alternately it could be relative to mankind's standard time measured by an atomic clock. This time value in spacetime is defined by the observer not the universe which has no built-in time value.

Therefore in relativity it is impossible to consistently relate any position or coordinate described by the observer to a physical point in the universe, using whatever geometry is proposed.
Newtonian physics always involved distances and times and never needed a geometry to transform between observation and reality because Newtonian physics needed no geometry in between to be manipulated.

Relativity defines a spacetime geometry so the observer can be given a distorted geometry with the rules defined by relativity not by Newtonian physics.

'
The mathematics of general relativity are complex. In Newton's theories of motion, an object's length and the rate at which time passes remain constant while the object accelerates, meaning that many problems in Newtonian mechanics may be solved by algebra alone. In relativity, however, an object's length and the rate at which time passes both change appreciably as the object's speed approaches the speed of light, meaning that more variables and more complicated mathematics are required to calculate the object's motion. As a result, relativity requires the use of concepts such as vectors, tensors, pseudotensors and curvilinear coordinates.
'
The critical statement:

Relativity is done by complicated mathematics.

From a Stanford document about spacetime:
'
Nevertheless, Einstein's theory of gravity represents a major swing back toward the relational view of space and time, in that it answers the objection of the ancient Stoics. Space and time do act on matter, by guiding the way it moves. And matter does act back on spacetime, by producing the curvature that we feel as gravity. Beyond that, matter can act on spacetime in a manner that is very much in the spirit of Mach's principle. Calculations by Hans Thirring (1888-1979), Josef Lense (1890-1985) and others have shown that a large rotating mass will "drag" an observer's inertial reference frame around with it. This is the phenomenon of frame-dragging, whose existence Gravity Probe B is designed to detect. The same calculations suggest that, if the entire contents of the universe were to rotate, our local inertial frame would undergo "perfect dragging" — that is, we would not notice it, because we would be rotating too! In that sense, general relativity is indeed nearly as relational as Mach might have wished. Some physicists (such as Julian Barbour) have gone further and asserted that general relativity is in fact perfectly Machian. If one goes beyond classical physics and into modern quantum field theory, then questions of absolute versus relational spacetime are rendered anachronistic by the fact that even "empty space" is populated by matter in the form of virtual particles, zero-point fields and more. Within the context of Einstein's universe, however, the majority view is perhaps best summed up as follows: Spacetime behaves relationally but exists absolutely.

If spacetime behaves only relationally (my interpretation) then it is connected only to that observer and it cannot exist absolutely since other observers cannot use the same spacetime.
Then it is also impossible for an observer on Earth to observe a distortion of a remote spacetime, like a black hole or a curvature to bend light near a distant mass.

Space time is always defined by the observer. There is no defined geometry for an absolute position in the universe so spacetime must be always relative to the observer.

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Last updated (08/25/2019)
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