"All of science is nothing more than refinement of everyday thinking", from "Physics and Reality", 1936  Albert Einstein ( 1879  1955 )
See: NASA's Gravity Probe B Confirms Two Einstein SpaceTime Theories
§ Why Riemanniann spacetime geometry?:
In Riemannian generalized curvilinear spacetime, not necessarily pseudo  Euclidean  space such as in locally Minkowski spacetime for special relativity, there are  affinely ( lt. affinis, meaning "connected with"; and in mathematics parallel lines and points are mapped to other related parallel lines and points by transformation laws ) connected spaces by intervening gravitational fields. And, therefore, for some vector field having vector components parallel ( gyroscopically ) transported around a given point along a given circuit curve, it can no longer be expected that in Riemannian generalized curvilinear spacetime the  components will return to their original positions unchanged: , vector field changed.
More succinctly, the "gravity ( Christoffel symbol ) affine" is the geometric proxy for the effects of gravity upon a parallel transported gyroscope!
The Riemann  Christoffel Curvature Tensor
( The Curvature Tensor )
Now we take the derivative in the opposite order by exchanging indices , so that we achieve differences in final position due to gravitational influences upon components of the vector field in their parallel ( gyroscopic ) transport along curve in transit about point ; and, therefore,
and then in the difference
of these two tensors, the following occurs:
This latter "modifying expression" proportional to vector field is the Riemann  Christoffel Curvature Tensor.
Hence by swapping indices we get
And because normally covariant derivations do not commute,
therefore
Permuting The Riemann  Christoffel Curvature Tensor:
The Riemann  Christoffel Curvature Tensor in a Geodesic ( locally pseudo  Euclidean Minkowski ) Coordinate Frame
§ Remember: There is a basic assumption throughout Einstein's General Relativity that any localized region of space  time becomes pseudo  Euclidean Minkowski and, hence, can be analyzed as a "patch" of flat ( special relativity ) Riemannian geometry around a chosen point.
Now, as before,
In a geodesic ( locally pseudo  Euclidean Minkowski ) coordinate frame around a point,
But the 2nd derivatives of the gravity potentials cannot all be eliminated in locally Minkowski ( pseudo  Euclidean ) coordinate systems around a point; and, hence, spacetime around is curved!
Therefore, assume non  zero
The Riemann  Christoffel Curvature Tensor Identities in a Geodesic Frame of Locally Minkowski Coordinates where
Summary: The Riemann  Christoffel Curvature Tensor in a Geodesic Frame of Locally Minkowski Coordinates where
Finally, since all these tensor equations are true in all geodesic ( locally Minkowski, pseudo  Euclidean coordinate systems ) frames, they are all true throughout the generalized Riemannian curvilinear spacetime geometry of general relativity.
The Bianchi Identities
We already know that the Riemann  Christoffel Curvature Tensor in a locally Minkowski, pseudo  Euclidean, coordinate system comprising a geodesic frame, reduces from
to
in the curved spacetime of general relativity.
Likewise in a general coordinate system, not locally Minkowski, the geodesic frame becomes
Therefore, we are left with
Now permuting the last 3 indices we get
And, hence, by the same tedious algebraic substitutions as before, we arrive at the following 5th rank tensors known as the Bianchi Identities:
These Bianchi 5th rank tensor identities are valid in all geodesic frames and hence are true throughout the general Riemannian curvilinear spacetime geometry of general relativity !
Einstein's Tensor: Preliminary Gravitational Field Equation
§ Method 1:
§ Method 2:
The Ricci ( or Contracted Curvature ) Tensor
By having twice contracted the Bianchi Identities, which itself is an expressed version of the Riemann  Christoffel curvature tensor, we have arrived at a considerably contracted curvature tensor  that is, everything else being equal!
Hence,
and in locally Minkowski ( pseudo  Euclidean ) coordinates where the Christoffels vanish,
Also, Ricci's Tensor is symmetric
which follows from
or,
The Invariant Curvature Scalar,
By contracting once more, the invariant curvature scalar is obtained:
or,
Of course, in a flat ( pseudo  Euclidean Minkowski ) special relativity spacetime metric, everywhere since all for constant throughout.
Two Grand Ideas are first needed: The Invariant Proper Volume and Gauss' ( Divergence ) Theorem
The Invariant Proper Volume
§ By Jacobi's Theorem:
§ The Invariant Volume Element:
Gauss' ( Divergence ) Theorem
Gauss' ( Divergence ) Theorem is a conservation of mass law valid in n  dimensional coordinate space which holds that the rate of net ( "sources" minus "sinks" ) outward flux of a vector field such as "energy  momentum" flowing thru a closed volume surface, is related to the boundary surface integral enclosing the volume's interior as follows:
§ In 3  dimensional space:
§ In 4  dimensional space, Gauss' Theorem applies with a volume element given as:
Derivation of the Einstein Gravitational Field Equations ( EFE ) from the Euler  Lagrange Equation of a Hamiltonian Variational Principle
§ Einstein's Gravity Equation in Simplest Terms:
And the rest of this derivation will be to show how to arrive at this equation!
§ The Hamilton Principle of Least ( Energy Expenditure ) Action:
In other words, it is not that one body of mass directly attracts another body of mass in the sense of Isaac Newton, but rather that bodies of mass deform the fabric of spacetime which in turn "pushes" matter according to which bodies of mass follow a path of least energy expenditure. Einstein's General Relativity mathematics therefore, interestingly and recursively, derived profoundly much from William Rowan Hamilton ( 1805  1865 )'s Least Action Principle! See: "Hamilton's Principle And The General Theory of Relativity", by A. Einstein, translated from "Hamiltonsches Princip und allgemeine Relativitätstheorie", found in "The Principle of Relativity", Dover Publications, Inc. For the original German in pdf click here and go to Einstein pages 1111  1116. [ note: you can directly type these pages or type in page '367' for page '1111'; type '372' for '1116'. Better: click here for the abreviated pdf, especially if the previous larger pdf does not download. ]
§ The Action Integral:
This "Action Integral" is structured so that its variations in coordinates and and their first partial derivatives as a body object transits from one possible path to another, are forced to ( a net ) zero by the Principle of Least ( Stationary ) Action for the least expenditure of transit energy.
That is,
which expresses the total variation according to the invariant Hamiltonian Variational Principle of the Action function for the gravitational field in the presence of matter and where
§ The Lagrange functions:
Now D. Hilbert and A. Einstein take the gravity Lagrange as
and the variation in the mass  energy Lagrange is subjected to the Euler  Lagrange function to finally arrive at a stress  energy ( or stress  energy  momentum, energy  momentum ) tensor designated as to be determined later.
§ Einstein's Gravity Field Equation ( EFE ):
which applies the concept that for celestial bodies in motion in gravitational spacetime, they follow curvilinear Riemannian paths of least expenditure of energy!
Exploring the Nature of the Stress  Energy  Momentum Tensor
Previously we discovered that throughout gravitational spacetime it is always possible to identify an irreducible, local Minkowski ( pseudo  Euclidean ) geodesic frame where the Christoffels vanish, , and for which
such that in special relativity mathematics, the differential equation for proper time
signifies the Lorentz invariant spacetime interval on any expanding "light sphere surface" as
Moreover and on closer examination of the EFE ( Einstein Field Equation ),
In any event, spacetime gravity is curvature in the sense that contrary to Newton where celestial masses "attract", celestial bodies are pushed by spacetime curvature along geodesic world timelines for the maximum proper transit time and the least expenditure of energy whereby escaping "gravity's grip" requires an extra expenditure of energy. See: Geodesic Spacetime Equation.
Now let's examine any 4 x 4 symmetric tensor as a matrix with components as
Hence, the total 16 components minus 6 top ( or an equal 6 bottom ) components gives 10 independent components!
But whenever we satisfy the condition of 4  spacetime coordinates, say as a given, then the final number of independent differential equations for the in the Einstein Field Equation ( EFE ) reduces down to 10  4 = 6!!
However we have assigned to the stress  energy  momentum density tensor the idea of sort of being like a fluid with a viscosity or thickness ( internal friction to flow movement ) as pictorially shown here:
§ Perfect ( idealized ) fluid:
In a "perfect fluid" which is isotropic ( equal in all directions ) in any comoving or rest frame, there is by definition zero viscosity as well as zero heat conduction all of which makes good intuitive sense. This in turn forces the "off diagonal" tensor components to be zero as well. Such an idealized fluid therefore is specified solely in terms of its "rest  frame energy density" and "rest  frame pressure" along the tensor trace as follows:
During the earliest times of the formation of the universe, when radiation dominated, this 4  energy  momentum velocity "perfect fluid tensor" best approximates the earliest primordial universe.
§ Dust:
This 4  energy  momentum velocity "dust tensor" best approximates the co  moving or rest frame of the universe during later times when radiation is negligible.
The Divergence of the Einstein Tensor
What actually confirms that Einstein's tensor with its ten variable components,
is the correct mathematical construction comporting identically well with the Lorentz invariant spacetime interval,
is the fact of the divergence of the Einstein tensor
which signifies that the geometric bending of spacetime fabric curvature is strictly and causally "attached" to the discrete mass of the body itself but not beyond!
But
or
in general curvilinear Riemannian metric space expressing the conservation of mass  energy in the presence of a gravitational field and thereby signifies:
1). matter is treated as continuous energy throughout gravitational spacetime;
2). the laws of energy and momentum are conserved for which the ordinary classical conservation laws of energy and momentum are easily derived from
Final Derivation of the Coupling Constantfor the Einstein Gravitational Field Equation ( EFE )
§ The mathematical physics tools we will be using:
§ Determining the coupling constant :
General Relativity Gravity while Deriving Newton's Law of Gravity using the Poisson Approximation Equation
Einstein's Field Equations ( EFE ) comprising General Relativity devolve to Newton's Law of Gravity in the corresponding limit for weak ( electromagnetic energy ) fields and in slow motions of relatively static distributions of matter.
Normally Newton's Law of Gravity is written as
and outstandingly Poisson's ( approximation ) equation links Einstein's gravity equation with Newton's gravity equation utilizing the proportional coupling constant in the following manner:
In simplier words, Newton's Universal Law of Gravity is an instantaneous "Action at a distance" force as a function of separating distances between two masses.
Summary: Poisson's Approximation Equation
Therefore Poisson's Equation is the linking bridge between Newton's and Einstein's worlds of gravity! Hence, see: "A derivation of Poisson's equation for gravitational potential", by Dr. Christian Salas.
§ References:
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