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Space Time Geometry

The observer's frame of reference uses a geometry. This geometry defines how the observer can describe his observations. His space time is not the physical universe. In relativity the space time is 'relative' to physical locations.

This distinction can be important.

(all quotes are from wikipedia)

In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum.

In relativity, spacetime is typically defined  with 3 dimensions in space plus time (your local time), so it is called space-time.  Spacetime is essentially a 4-D geometry with 4 coordinates. 'Relativity' involves your frame of reference 'relative' to the physical space. Time is always local but its zero reference is defined as needed.
 The 3 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.

Some propose the big bang started time. That is wrong. The observer defines the time 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.

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 typical X0,Y0,Z0  is at the lower left corner in the observer's frame of reference.

Relativity assumes the location in the observer's observer's frame of reference are described in the Euclidean geometry with XYZ. Distances are calculated from changes in position.

Relativity is about the observer in acceleration. A non-accelerating observer follows normal physics. Therefore the spacetime for relativity effects is about distances being moved while accelerating not the actual position of the observer.

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.

Other 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 GPS.

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

Astronomers use an equatorial coordinate system (ECS).
The equatorial coordinate system 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.

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 a zero reference also defined for each of those planes. Distances to objects would be relative to that universe zero reference point.

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

By definition the universe is everything.
There cannot be two universes, by the very definition of the word.

Some theorists propose multiple universes.
This theory is definitely not multiple geometries. A multiverse definition is clear.

This is just a fantasy. It takes an observer (like you or me) to describe our universe.
To propose an external universe with no possible observer is not logical. There is no difference between no external universe and an invisible universe with no observer to detect it . It is not logical to propose something like a universe as invisible.
If an object is proposed to go to this invisible universe and yet it returns changed physically  then obviously the theory is wrong. The change had to occur in this universe; the other universe is always a fantasy, with no logical justification. If I toss a dry ball high in the air but when I catch it now it is wet,  it is utterly laughable to claim it got wet while it was in the parallel universe. An alternate universe is always a fantasy.

Some cosmologists propose special geometries.

In general relativity Kruskal–Szekeres coordinates, named after Martin Kruskal and George Szekeres, are a coordinate system for the Schwarzschild geometry for a black hole. These coordinates have the advantage that they cover the entire spacetime manifold of the maximally extended Schwarzschild solution and are well-behaved everywhere outside the physical singularity.

This definition of a geometry for the observer at the black hole is confused.
The singularity is a proposed phenomenon in the observer's spacetime and is definitely not a physical entity.

In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed.

Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space.

A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike.

In differential geometry, a differentiable manifold is a space which is locally similar to a Euclidean space. In an n-dimensional Euclidean space any point can be specified by n real numbers. These are called the coordinates of the point.

An n-dimensional differentiable manifold is a generalisation of n-dimensional Euclidean space. In a manifold it may only be possible to define coordinates locally. This is achieved by defining coordinate patches: subsets of the manifold which can be mapped into n-dimensional Euclidean space.

When a surface has a constant zero Gaussian curvature, then it is a developable surface and the geometry of the surface is Euclidean geometry.

When a surface has a constant positive Gaussian curvature, then it is a sphere and the geometry of the surface is spherical geometry.

When a surface has a constant negative Gaussian curvature, then it is a pseudospherical surface and the geometry of the surface is hyperbolic geometry.

All of these theoretical gyrations involve the observer's frame of reference and a proposed geometry to help describe the theory.

Each model, manifold, or bridge, or curvature is just theoretical for the observer's geometry but not a physical behavior.

Relativity is about handling distances moved  in 3-D over time not actual positions.

Quantum physicists propose a complex geometry for spacetime (with 11 or more dimensions) so this geometry might help describe the behaviors at the subatomic level.
Space does have any dimensions 'built-in' but the observer defines the geometry to use.

Spacetime is always the observer's frame of reference.

Date updated 03/24/2019

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