To have peaceful alien contact , the space time of our sense and perceptions are related to aliens also having such space time functions that either diverge or converge.
Spacetime split. In spacetime algebra, a spacetime split is a projection from four-dimensional space into (3+1)-dimensional space with a chosen reference frame by means of the following two operations: a collapse of the chosen time axis, yielding a 3D space spanned by bivectors, and.
Spacetime split[edit]Spacetime split – examples:
{\displaystyle x\gamma _{0}=x^{0}+\mathbf {x} }
{\displaystyle p\gamma _{0}=E+\mathbf {p} }[1]
{\displaystyle v\gamma _{0}=\gamma (1+\mathbf {v} )}[1]
where {\displaystyle \gamma } is the Lorentz factor
{\displaystyle \nabla \gamma _{0}=\partial _{t}-\nabla }[2]In spacetime algebra, a spacetime split is a projection from four-dimensional space into (3+1)-dimensional space with a chosen reference frame by means of the following two operations:
- a collapse of the chosen time axis, yielding a 3D space spanned by bivectors, and
- a projection of the 4D space onto the chosen time axis, yielding a 1D space of scalars.[3]
{\displaystyle {\begin{aligned}x\gamma _{0}&=x^{0}+x^{k}\gamma _{k}\gamma _{0}\\\gamma _{0}x&=x^{0}-x^{k}\gamma _{k}\gamma _{0}\end{aligned}}}As these bivectors {\displaystyle \gamma _{k}\gamma _{0}} square to unity, they serve as a spatial basis. Utilizing the Pauli matrix notation, these are written {\displaystyle \sigma _{k}=\gamma _{k}\gamma _{0}}. Spatial vectors in STA are denoted in boldface; then with {\displaystyle \mathbf {x} =x^{k}\sigma _{k}}the {\displaystyle \gamma _{0}}-spacetime split {\displaystyle x\gamma _{0}} and its reverse {\displaystyle \gamma _{0}x} are:
{\displaystyle {\begin{aligned}x\gamma _{0}&=x^{0}+x^{k}\sigma _{k}=x^{0}+\mathbf {x} \\\gamma _{0}x&=x^{0}-x^{k}\sigma _{k}=x^{0}-\mathbf {x} \end{aligned}}}
find more on:
https://en.wikipedia.org/wiki/Spacetime_algebra
In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), ... is theMinkowski metric with signature (+ − − −). Thus .... In spacetime algebra, a spacetime split is a projection from four-dimensional space into ...
https://www.jstor.org/stable/2159148
Minkowski space is space-like if the induced metric is positive definite. Such a surface has two S2-valued Gauss maps. The notion of split curvature measures.
image credit:
crev.info
from
Inspire-hel
Inspire-hel
In this image we see a portion of 33-dimensional Minkowski spacetime embedded in Euclidean 33-space. The accelerated observer γγ is indicated by a curved line going upwards. At three different instances τ1,τ2,τ3τ1,τ2,τ3 of γγ's proper time, a part of the respective past light cone is shown. Since the spacetime is flat, the cones do not intersect themselves. Lines of constant radial distance, as measured by the observer, are schematically drawn in accordance with the direction of γγ's tangent vector. We remind the reader that orthogonality in Minkowski space differs from orthogonality in Euclidean space.
credit:
http://inspirehep.net/record/1654928
2019
february 8
Abstract (arXiv)
We motivate and construct a mathematical theory for the separation of space and time in general relativity. The formalism only requires a single observer and an optional choice of reference frame at each instant. As the splitting is done via the observer's past light cone, it is both closer to the experimental situation and mathematically less restrictive than the splitting via observer vector fields or spacelike hypersurfaces. Indeed, the theory can in principle be applied to all spacetimes and adapted to other `metric' theories of gravity. Instructive examples are developed along with the general theory. In particular, we obtain an alternative description for accelerated frames of reference in Minkowski spacetime. Further, we use the splitting formalism to motivate a new mathematical approach to the Newtonian limit of the motion of mass points. This employs a general formula for their observed motion, distinguishing between `actual' forces (i.e. those detectable via an accelerometer) and pseudo-forces. Via this formula we show that for inertial frames of reference in Minkowski spacetime the essential laws of non-gravitational Newtonian mechanics can be derived. Physically relevant, related, open problems are indicated throughout the text. These include the proof, that the Newtonian limit gives rise to the central pseudo-forces known from Newtonian mechanics (`constant gravity', Euler, Coriolis and centrifugal force) for non-inertial frames of reference in Minkowski spacetime, as well as the derivation of Newton's law of gravitation in the Schwarzschild spacetime under said limit. This is a slightly corrected version of a master's thesis in mathematical relativity, written at TU Berlin in 2016/2017. Comments by the reviewers have been taken into account. If there are any remaining errors, they are solely due to the author.
Note: * Brief entry *
Note: 109 pages on A4 paper; 3 figures; CC BY-NC 4.0 license; keywords: space-time splitting, Newtonian limit, relativistic kinematics, frame of reference, gravitational lensing
image credit
tallwhitealiens.
The comparison is made with the Minkowski space.
http://inspirehep.net/record/1654928
2019
february 8
Abstract (arXiv)
We motivate and construct a mathematical theory for the separation of space and time in general relativity. The formalism only requires a single observer and an optional choice of reference frame at each instant. As the splitting is done via the observer's past light cone, it is both closer to the experimental situation and mathematically less restrictive than the splitting via observer vector fields or spacelike hypersurfaces. Indeed, the theory can in principle be applied to all spacetimes and adapted to other `metric' theories of gravity. Instructive examples are developed along with the general theory. In particular, we obtain an alternative description for accelerated frames of reference in Minkowski spacetime. Further, we use the splitting formalism to motivate a new mathematical approach to the Newtonian limit of the motion of mass points. This employs a general formula for their observed motion, distinguishing between `actual' forces (i.e. those detectable via an accelerometer) and pseudo-forces. Via this formula we show that for inertial frames of reference in Minkowski spacetime the essential laws of non-gravitational Newtonian mechanics can be derived. Physically relevant, related, open problems are indicated throughout the text. These include the proof, that the Newtonian limit gives rise to the central pseudo-forces known from Newtonian mechanics (`constant gravity', Euler, Coriolis and centrifugal force) for non-inertial frames of reference in Minkowski spacetime, as well as the derivation of Newton's law of gravitation in the Schwarzschild spacetime under said limit. This is a slightly corrected version of a master's thesis in mathematical relativity, written at TU Berlin in 2016/2017. Comments by the reviewers have been taken into account. If there are any remaining errors, they are solely due to the author.
Note: * Brief entry *
Note: 109 pages on A4 paper; 3 figures; CC BY-NC 4.0 license; keywords: space-time splitting, Newtonian limit, relativistic kinematics, frame of reference, gravitational lensing
image credit
tallwhitealiens.
The comparison is made with the Minkowski space.
The picture shows a typical domain of the static observer mapping in 33-spacetimes. We chose to transform the coordinates (???)(???) to polar coordinates (r,ϕ)(r,ϕ). The origin is accentuated by the ring in the middle, the straight lines diverging away are (ϕ=\const)(ϕ=\const)-lines, the circles are (r=\const)(r=\const)-lines, both in equal spacings. The outer dotted line indicates the region where the mapping is undefined.
The picture shows a typical domain of the static observer mapping in 33-spacetimes. We chose to transform the coordinates (???)(???) to polar coordinates (r,ϕ)(r,ϕ). The origin is accentuated by the ring in the middle, the straight lines diverging away are (ϕ=\const)(ϕ=\const)-lines, the circles are (r=\const)(r=\const)-lines, both in equal spacings. The outer dotted line indicates the region where the mapping is undefined.
The picture indicates the measurement of distances on the past tangent light cone c−qcq− at a point qq in a three dimensional spacetime by embedding the tangent space \CapTq\spti\CapTq\spti into Euclidean 33-space. At the top we see the tangent vector of the observer at some fixed time, which gives rise to an orthogonal hyperplane. The cone c−qcq− is situated below. The surface of constant radial distance rr on c−qcq−, as measured by the observer, is orthonally projected to a circle of radius rr on the hyperplane. The observer's measurement of angles is analogous.