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Einstein's ( EFE ) Gravitational Field Equation

"All of science is nothing more than refinement of everyday thinking", from "Physics and Reality", 1936 - Albert Einstein ( 1879 - 1955 )

§ 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:

1. the partial derivatives of the covariant vector field will cancel;

2. and, after canceling several other terms, the remaining as a function of the fundamental metric tensor representing gravitational potentials, will be proportional to and will modify ( bend, twist, and warp ) vector field comprising spacetime fabric.

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 co-moving 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

• Poisson's Approximation Equation is the unique solution to Einstein's General Relativity Gravity Equation;

• Poisson's Approximation Equation is the unique underlying equation which both generates Newton's 2nd Law of Motion as well as Newton's Universal Law of Gravity.

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.

 Dutch physicist Hendrik Antoon Lorentz (1853 - 1928) photographed with Einstein in Leiden in 1921 Sir Isaac Newton ( 1643 – 1727 )Portrait by Godfrey Kneller in 1689 - Newton at age 46 Einstein at the blackboard during his 1922 lecture at the Sorbonne in Paris

§ References:

1. "The Meaning of Relativity", by A. Einstein, Dover Publications, Inc.

2. "Hilbert's 'World Equations' and His Vision of a Unified Science", by U. Majer and T. Sauer

3. "Die Grundlagen der Physik", by David Hilbert, 1915

4. "New research on the discovery of the field equations of general relativity by David Hilbert and Albert Einstein", by Daniela Wünsch, Institute für Wissenschaftsgeschichte, Göttingen, Germany

5. "A derivation of Poisson's equation for gravitational potential", by Dr. Christian Salas

6. "Introduction to Tensor Calculus, Relativity and Cosmology", by D. F. Lawden, Emeritus Professor, University of Aston in Birmingham, U.K., Dover Publications, Inc.

7. "Introduction to the Theory of Relativity, with Foreword by Albert Einstein", by Peter Gabriel Bergmann, Dover Publications, Inc.

8. "The Theory of Relativity", 2nd edition, by R.K. Pathria, Distinguished Professor Emeritus University of Waterloo, Dover Publications, Inc.

9. "Tensors, Relativity and Cosmology", by Mirjana Dalarsson, Ericsson Research and Development and Nils Dalarsson, Royal Institute of Technology, Stockholm, Sweden, Elsevier Academic Press

10. "Relativity: Modern Large - Scale Spacetime Structure of the Cosmos", Editor Moshe Carmeli, formerely of Ben Gurion University, Israel, World Scientific Publishing Co. Pte. Ltd.

11.  "Hamiltonsches Princip und allgemeine Relativitätstheorie", von A. Einstein, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften ( Berlin, 26, Oktober 1916 ), Seite 1111-1116, original German, now in archive at the Library of Congress: for the pdf go here and 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.

12.  "Méthodes de calcul différentiel absolu et leurs applications", by Ricci and Levi-Civita, 1900

13.  "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?", by A. Einstein, B. Podolsky, and N. Rosen, Institute for Advanced Study, Princeton, Physical Review, Vol. 47, May 15, 1935. This is a seminal paper in the history of quantum physics and Heisenberg's Uncertainty Principle.

14.  "The Particle Problem in the General Theory of Relativity", by A. Einstein and N. Rosen, Institute for Advanced Study, Princeton, Physical Review, Vol. 48, July 1, 1935. Seminal companion to the May 15th published paper in the history of quantum physics and Heisenberg's Uncertainty Principle.

15.  "The Evolution of Scientific Thought - From Newton to Einstein", by A. d'Abro, 1st edition © 1927; 2nd edition © 1950. Wonderful exposition of the history of scientific thought from the pre-relativity physics of Galileo and Sir Isaac Newton to a brief discussion of the expanding universe since the big bang of Belgian Abbé Lemaitre as a consequence of Albert Einstein's newer theory of General Relativity introduced in 1916.

16.  "Introduction to General Relativity", by Gerard't Hooft, Institute for Theoretical Physics, Utrecht University and Spinoza Institute, December 2012

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