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Space time Geometry

Much of modern physics involves the geometry of spacetime.
Some times this geometry is interpreted wrong; this is a long justification for that conclusion because I expect some disagreement.

First - explaining a geometry

Every person has their view of the universe, called their frame of reference.
To describe positions of anything in that view a geometry is required. The geometry defines the relationship of the axes.

The observer's frame of reference uses the user's selected geometry. This geometry defines how the observer can describe his observations.
There are many geometries available.
The simplest is Euclidean geometry with its 3 axis coordinates for left/right, up/down, in/out; these are typically designated by the 3 letters X,Y,Z. The observer defines the zero reference for each axis coordinate. The X0,Y0,Z0  is often at the lower left corner in the observer's frame of reference.
This zero reference is usually adjusted by the observer.
Time is sometimes used as a 4th coordinate.

If I tell another observer of my observation at X1Y2Z3,T5 the other observer must use the same geometry to understand those coordinates.
If the other observer is 1 meter to the right his observed X coordinate is different; but he can offset that X coordinate to view the same observation for that expected behavior but in his local frame of reference. For correct observations the two observers must use the same geometry and their correct zero reference.

This combination is critical.  With the same geometry and its zero reference both observers are observing the same space defined by this geometry.

For multiple observers to interpret the same observed behavior from their individual frames of reference they must share the same geometry and its zero reference.

There are other common geometries.

GPS devices use a geographic coordinate system.

A geographic coordinate system is a coordinate system that enables every location on the Earth to be specified by a set of numbers or letters, or symbols.  A common choice of coordinates is latitude, longitude and elevation.
We agree on the zero reference for each axis in the GPS system.

The observer can use mutiple non-conflicting geometries. For example the GPS can define the current physical location on Earth while also using a small Euclidean space to describe 4-D motions within a smaller frame.

Note this geometry is not part of the Earth; it is used by the observer.

An observer on Mars could interpret an observed event on Earth correctly when both observers use the same GPS geometry and its zero reference, like an observation of a volcano near Alaska.

Astronomers use an equatorial coordinate system (ECS).
The ECS is a celestial coordinate system widely used to specify the positions of celestial objects. It may be implemented in spherical or rectangular coordinates, both defined by an origin at the centre of Earth, a fundamental plane consisting of the projection of Earth's equator onto the celestial sphere (forming the celestial equator), a primary direction towards the vernal equinox, and a right-handed convention.

The origin at the center of Earth means the coordinates are geocentric, that is, as seen from the centre of Earth as if it were transparent. The fundamental plane and the primary direction mean that the coordinate system, while aligned with Earth's equator and pole, does not rotate with the Earth, but remains relatively fixed against the background stars. A right-handed convention means that coordinates increase northward from and eastward around the fundamental plane.

Along with the distance from the Earth to the object, the ECS enables a description for the location of every object in the observable universe.

Every observer on Earth using the same ECS geometry and its zero reference  can share observations with any other observer on Earth.

Note this ECS geometry is not part of the universe but is used by the observer.
To describe a location of objects in a context not geocentric a new geometry must be defined.

The simplest 'universal' geometry is an adaptation of the ECS.
A galaxy that seems to be stationary relative to all its neighbors could be assigned the 'universe zero reference' and in relation to that point a right ascension plane and a declination plane could be defined for the universe, and also a zero reference could be defined for each of those planes, using that selected galaxy. Distances to objects would be relative to that universe zero reference point.

This theoretical universal coordinate system (UCS) enables the observer to describe the location for any object in the entire universe, whether visible or not.

Note this UCS geometry is not part of the universe but is used by the observer.
This UCS is not defined yet so it cannot be used.

Therefore the best that can be done now is an Earth-centered geometry.

It is currently impossible to describe a position in the universe other than by the observer's frame of reference.

This is an important limitation for some theories.

If a theory proposes a fabric of space that description cannot use any 'fixed' positions in the universe like for an embedded fabric with defined edges.

Explaining - Relativity space time geometry

Relativity assumes the location in the observer's observer's frame of reference are described in the Euclidean geometry with 3 axes, XYZ. Distances are calculated from changes in position during the time of acceleration.

Relativity is about the observer in acceleration. A non-accelerating observer follows normal physics.
The distortion in the observer's frame of reference is called space time curvature. Therefore a second, non-accelerating, observer cannot  interpret the frame of reference of the first observer without a common frame of reference - or a common space time. The first observer has a distorted space time but the second cannot have an identical distorted space time.
They cannot share an observation.

The famous twin paradox for relativity describes this difference in frame of reference between the two observers. Each observer cannot see the other's space time.

This restriction is important.
There are wrong interpretations when ignoring this restriction.

If the accelerating observer encounters a huge mass that supposedly collapses into a singularity no other observer can share in that observation.
By the theory of relativity only an accelerating observer right there can observe a black hole.
By the theory of relativity no one else in the universe can observe that specific  black hole.

Cosmologists propose millions of various sized black holes are seen in the universe but that proposal violates the theory of relativity that is used to claim such a thing could exist.

Some people interpret spacetime curvature as a static phenomenon observable by everyone. An example of this is proposing the universe's space time is curved around a distant galaxy. That proposal for all observers violates the theory of relativity.

This space time is not the physical universe; neither is the first observer's curvature. In relativity the observer's space time (and its curvature) is 'relative' to physical locations.

This distinction can be important.
 The dimensions for describing a location in space are defined in the observer's geometry. Space has no built-in geometry.

Some propose space has a built-in geometry, or one was created by the big bang. That is wrong. The observer defines the geometry being used, not the universe.

Some propose the big bang started time. That is wrong. The observer defines the time coordinate being used. Some suggest using cosmological time; its zero reference is the big bang event. Even then that time definition is part of the observer's frame of reference and is not actually part of the universe. In practice we use a standard 'universal' time maintained by the counts of an atomic clock. An observer can use that time for observations or the zero reference for his time coordinate can be changed. One simple example is to offset the 'universal' time coordinate by the current time at the start of observations. This offset defines time=0 at the beginning and then the time coordinate during observations is the amount of time since the observation began.
Sometimes cosmologists prefer to think of time=0 at the big bang event.

This is done by using the theoretical amount of time since the big bang as the time coordinate offset; this calculated time since the big bang is called cosmological time.

In relativity, spacetime is typically defined  with 3 dimensions in space plus time (your local time), so it is called space-time. 

From an Einstein equation description:
The stress–energy tensor involves the use of superscripted variables (not exponents; see tensor index notation and Einstein summation notation). If Cartesian coordinates in SI units are used, then the components of the position four-vector are given by: x0 = t, x1 = x, x2 = y, and x3 = z, where t is time in seconds, and x, y, and z are distances in meters.

Note relativity uses distances from the current positions in a Euclidean geometry.

Relativity and its spacetime never reference an absolute position in the universe; this spacetime is based the observer's current position.

One notable feature of string theories is that these theories require extra dimensions of spacetime for their mathematical consistency. In bosonic string theory, spacetime is 26-dimensional, while in superstring theory it is 10-dimensional, and in M-theory it is 11-dimensional. In order to describe real physical phenomena using string theory, one must therefore imagine scenarios in which these extra dimensions would not be observed in experiments.

This geometry involving many more dimensions is apparently required for the observers of string theory behaviors to describe their observations.

This spacetime geometry with many dimensions is not part of the universe; it is used by the accelerating observer. Multiple observers would share this geometry.

I have encountered some cosmology theories that describe a geometry for the fabric of space. The universe follows or conforms to this geometry.
I expect that theory to be impossible. It is impossible to describe physical locations and behaviors at the level of the entire universe when restricted to the observer's frame of reference.

Spacetime is always the observer's frame of reference.
It is never physical space observed by all.

There is another consequence for cosmology in its use of space time in addition to no black holes (mentioned above): no distant curvature.

Gravitational lensing is proposed for light bending around distant objects. Here on Earth we cannot observe a distant spacetime and its curvature.

Light is bent whenever the width of the beam passes through different densities of matter. A prism or the surface of water demonstrate this behavior. An atmosphere is densest at the surface due to gravity on the gas. The 1919 observation of a star's position being bent by gravity was due to checking the light at the Sun's surface; this is at the bottom of the Sun's atmosphere. A true test of gravitational lensing should have been done above the atmosphere, but it was not.

I have seen online claims other planets and stars (other than Regulus) did not all bend as expected by Sun's gravity alone in 1919 but I know nothing about their accuracy.
Gravitational lensing is used by cosmologists to address an object observed in a wrong place by claiming that observation is an illusion caused by the light bending to appear where it should not be.
The typical scenario is where a high red shift quasar is observed in front of or next to a low red shift object. This is claimed to be an illusion. I posted about the quasar red shift a few days ago.

If this phenomenon were real there should be many illusions around every huge galaxy; M31 does not show any.

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