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A Derivation of a Motion Sensitive Einstein Tensor for Contextualized Rotating Mass
eBook (PDF): $6.95
Location vectors that describe the positions of points on spheres that are centered at the origin of coordinates can be constrained to specify rings that rotate about their axis of symmetry with an... More > angular velocity that is characteristic of the ring. When the partial derivatives of said vectors are put into linear combinations that produce at every point in space-time a system of four mutually perpendicular unit length reference vectors those pointwise systems may aggregate to comprise for space-time a vector field of orthonormal reference frames. We can devise a simple system for which the Riemann tensor contains no explicit dependence on the angular velocities, or a relatively complicated system for which they do show up in Riemann tensor components. But the physical quantities of the stress-energy-momentum tensor components are specific to a vector reference system of choice, so we calculate a Riemann tensor that suits that choice and then the Einstein tensor that suits the system.< Less
A Completely Differentiated Rotation Vector Field
eBook (PDF): $1.55
Partial differentiation and tensor product space algebra are combined to produce fascinating results from vector and dual spaces. The mathematical constructs which emerge are then given an... More > application in a space that fills the image of a four dimensional bijective map with space-time points that are in a state of rotation about a single axis.< Less
Axially Rotating Vector Frames Completely Differentiated In Four Dimensional Space-time
eBook (PDF): $3.77
Functions that multiply the vectors we obtain by differentiating space-time points partially with respect to scalar parameters that are in a 1-to-1 and onto relationship with the points may change... More > the size we associate with a given direction, but not the direction, so that by changing the coefficient functions we change the parallelepiped shapes associated with each origination of four different directions at each space-time point. Thusly do derivatives of products made between basis vectors and their coefficient functions become of geometric interest. Moreover, a 1-to-1 and onto relationship between space-time points and their four corresponding parametric scalar values connotes that the latter can be differentiated by means of gradients taken with respect to the coordinates of the former, so that meaningful shape changing in space-time nurtures expectations for complementary effects being useful to the regulation of the dual space that the gradients generate. These issues are explored herein.< Less
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