The Geodesic Equation for Body Motion in Gravitational Fields

"The eternal mystery of the world is its comprehensibility" - ( 1879 - 1955 )

§ Einstein's "Law of Equivalence of Acceleration and Gravity" or "The Principle of Equivalence" :

Fields of acceleration and fields of gravity are equivalent physical phenomenon. That is, if there's no "upward" acceleration, then there's no "downward" gravity.

Preliminary Understanding of the following Mathematical Symbols

Naïve realism easily allows ordinary understanding of the mathematical symbols of ; however, for deeper understanding of physical reality it becomes necessary to invent newer and more powerful mathematical tools such as the Christoffel symbol by which a deeper probing of external reality is made entirely possible. Hence, a certain amount of patient indulgence is asked of the serious reader of this mathematical essay since it is strongly suggested that by a thorough reading of these equations and the tightly woven logic by which they are presented, that with final patience the reader of Relativity Calculator will have a fairly good grasp of the underlying mathematical of Einstein's ultimate gravitational equation!

A Quick Introductory Meaning of Geodesic Motion

Sir Arthur Eddington ( 1882 - 1944 ) on May 29, 1919 tentatively, but empirically, confirmed Einstein's mathematical general relativity conjecture that light passing near the gravity field of the sun was slightly, but convincingly, bent in the spacetime proximate vicinity of the sun, thus showing that on grand cosmic scales there is no such reality for Euclid's straight geometry lines. Likewise, gyroscopic motion will equivalently follow the curvilinear contours of spacetime fabric as influenced by other gravitational fields. In fact, the very best way to "map out" the contours of gravity's spacetime fabric is by setting gyroscopes aloft, so to speak, into the grand cosmos where these gyroscopes will follow the same transit paths by their parallel ( gyroscopic ) transport as their cousin light beams travelling the exact same intervening gravity fields.

Naturally, as these gyroscope cousins of light beams trace out spacetime's fabric, their "spin - time" as well as "transit - time" of passage will equivalently also be effected by gravity's field; thus, we arrive at space - time!

Hence, the meaning of "geodetic motion" is best understood in terms of the parallel transport of gyroscopes! And the mathematics for such understanding was arrived at in 1854 by Bernhard Riemann in his doctoral "On the Hypothesis That Lie at the Foundations of Geometry", German version; where, the derivation of such geodesic equation for motion in gravity fields is therefore the purpose of this following mathematical exposition:

Relativity Science Calculator A Quick Summary for the Geodesic Equation In Riemannian geometry for space as opposed to flat Euclidean metric space, the "shortest curve" or "straightest" possible world timeline with minimum actionable variation in curved spacetime according to the Principle of Least Action between two given fixed points and is determined by the set of solutions to the "timelike" geodesic differential equations, the optimal one of which gives a "stationary" or minimal value to the calculus action integral for the extremal maximum proper transit time for the most efficient use of energy:

Relativity Science Calculator Recapitulating

A true ( idealized ) geodesic motion for a body object is where a "timelike" world timeline parallel transports, according to the Principle of Least Action, its own tangent vector along the "most consistently stationary" curvilinear spacetime coordinates while maintaining constant magnitude and direction for which

Relativity Science Calculator That is, to explicate further, suppose:

Relativity Science Calculator NASA's Gravity Probe B Confirms Two Einstein Space-Time Theories

See: NASA's Gravity Probe B Confirms Two of Einstein's Space - Time Theories

Relativity Science Calculator In other words, the geodesic world timeline is that gyroscopic ( parallel transport ) transit path for any object body or sub - atomic test particle which expends the least amount of energy according to the Principle of Least Action!

Hence in Euclidean flat space the shortest distance between fixed points and is exactly equivalent to the shortest world timeline transit time; but in Riemannian curved geometric spacetime with an intervening gravitational mass such as earth's sun within our solar system affecting velocities, it therefore becomes the extremal maximum proper time ( and not minimum straight - line distance! ) which determines the geodesic world timeline connecting fixed points and .

Now in order to achieve the derivation for the geodesic spacetime equation describing the minimum world timeline between fixed spacetime points and , we must therefore develop several new "mathematical technologies" as follows:

Several New Mathematical Technologies

1). Parallel ( Gyroscopic ) Transport provides the underlying conceptual basis for the Christoffel symbol; additionally, parallel ( gyroscopic ) transport gives an immediate derivation of the geodesic equation which at this time in our mathematical essay will be deferred until later for a deeper understanding of all of these new mathematical constructs;

2). The Christoffel symbol accounts for the rate of change of the local basis ( or its reciprocal ) which varies from infinitesimal world timeline spacetime point to infinitesimal spacetime point in the affine ( affinity ) geometry of curved Riemannian space. In effect, therefore, the Christoffel symbol mathematically represents the intensity of the composite gravitational force field under conditions of parallel ( gyroscopic ) transport;

Relativity Science Calculator 3). The non - tensor Christoffel symbol is "arithmetized" by the spacetime gravitational potentials , 2nd - order metric tensor, of the underlying curvilinear coordinates whose tensor proof is proffered later on;

4). The Euler - Lagrange ( or Euler's, Lagrange's ) equation of motion:

whose derivation will later on also be given.

Parallel ( Gyroscopic ) Transport in an Affinely Connected Parallel Space

Demonstrating Gyroscopic Parallel Transport in the International Space Station using "gravity pushing" straws simulating parallel transport - i.e., gravity's parallel transport influence upon a gyroscope

§ Vector Analysis in terms of Covariant and Contravariant component expansion:

Relativity Science Calculator § Vector Analysis in terms of differential calculus:

§ Vector Analysis in terms of partial differentials: Now the partial differential component expansion for the total differential is given as follows:

§ Derivation of The Law of Parallel Displacement for Parallel ( Gyroscopic ) Transport: Employing a homogeneous ( constant in magnitude and direction ) vector field , we now consider its parallel displacement between coordinate points to ( using the variational calculus operator - not Kronecker's delta! ) signifying an infinitesimal parallel vector movement along a given - axis for some infinitesimal parallel translation of vector as follows:

Therefore,

Relativity Science Calculator So, the law of parallel displacement simply becomes

Or, equivalently, but only using the components of vector and only using the upper index in for simplicity's sake, we derive and hence also define the following:

Relativity Science Calculator However, a simpler and more intuitive parallel transport derivation for a vector field which is homogeneously constant in both magnitude and direction is immediately given as follows:

where, nevertheless, the following two terms are missing

all of which means that in parallel ( gyroscopic ) transport all vector field component changes ( magnitude, direction ), for whatever "parallel direction" ( or lateral push of the gyroscope ) is invoked, are therefore null or equivalently set to zero which is what we would naïvely expect.

Derivation of The Covariant Derivative of the Vector under the Condition of Parallel ( Gyroscopic ) Transport

Now that the law of parallel ( gyroscopic ) displacement has been established, namely

where the covariant component expansion of vector is parallel transported to along a given - axis which in turn generates a parallel translated vector at coordinate . Therefore,

Relativity Science Calculator is some unknown, yet still to be determined "4 x 4 x 4 - component object", system of functions of local ( or its reciprocal ) base vectors. That is, is some arbitrary system of differential functions that characterizes spacetime transport according to the law of parallel displacement and which therefore numerically represents the rate of change of the local ( or its reciprocal ) basis whenever a test particle or object body transits the extremal minimum geodesic path thru intervening gravitational fields for the least action and most efficient expenditure of energy.

Furthermore, is linear in both as well as in since the vector sum must also parallel transport according to the same transformation law as for each of the individual vectors.

Therefore, these mathematical objects which are dependent on the particular spacetime coordinate system in which they arise, will hence go to null in Euclidean space; that is, in Descartes coordinates.

Whereas

defines the parallel transport of a covariant vector according to the law of parallel displacement, so also there must analogously be the parallel transport of a contravariant vector under the condition that the scalar composition between these two absolute vectors cannot change under parallel transport. That is,

And, therefore, an analogous construction utilizing the parallel transport for contravariant vector provides:

Relativity Science Calculator Finally, notice that in Euclidean Descartes coordinate space where , the covariant derivatives of either covariant and / or contravariant vectors reduce down to their respective ordinary partial derivatives!

Tensor Rules for Higher - Order Construction of Covariant Derivatives

Rule: The number of tensor terms equals the number of tensor indices:

Examples:

Relativity Science Calculator Demonstrating that the Christoffel Symbol Represents the Rate of Change of the Local Basis Under Conditions of Parallel ( Gyroscopic ) Transport

We already know that

Hence, the Christoffel symbol of the 2nd - kind with 27 ( ) components becomes, in terms of the local basis vectors, as follows:

Thus, the covariant derivative of a homogeneous vector field in Euclidean space devolves to ordinary partial derivatives where all Christoffel symbols of the 2nd - kind vanish - i.e., . What is further being shown here is that the covariant derivative of a homogeneous vector field involves not only the rate of change of the vector field itself but also the rate of change of the local basis as a body moves thru the "gravity effects" of Riemannian curvilinear space from spacetime point to spacetime point. In effect, therefore, the Christoffel symbol mathematically represents the intensity of the composite gravitational force field under conditions of parallel ( gyroscopic ) transport whereby an infinite number of gravitational potentials on the curvilinear spacetime fabric surface "arithmetize" the Christoffel !

Parallel (Gyroscopic ) Transport Summary

Relativity Science Calculator The Christoffel Symbols:

§ as an operator raising and lowering indices:

Recapitulating the following definitional relationships ( from earlier above ):

and now define the 2nd - order metric tensor of the underlying curvilinear coordinate system

for which

forms a unit symmetric system; therefore the following derivations are true:

Or, relating the covariant and contravariant components of the same 1st - order tensor ( vector ):

Likewise, relating the covariant and contravariant components of the same 1st - order tensor ( vector ):

In fact, relating the same covariant ( tensor ) derivative of both covariant and contravariant vector components, we easily get:

Relativity Science Calculator Whereas, always, the and are the respective covariant and contravariant projections of vector onto the coordinate axes. Notice that the differential vector equation

can, for the , be analogously be written as

and, hence, we are moving towards a more generalized geometry in the following

whereas, the invariant spacetime interval ( see: 'Proper Time' ) is written as

Then further define ( see above ) the 2nd - order metric tensor as

Relativity Science Calculator Furthermore,

In fact, the following relations are entirely true:

Relativity Science Calculator Hence,

§ Proof of the interrelations of the 1st & 2nd kind Christoffel symbols:

Relativity Science Calculator § The relation between the Christoffel symbol and the metric tensor of the underlying curvilinear spacetime coordinate system:

The relationship between the Christoffel symbol and the metric tensor of the underlying curvilinear spacetime coordinate system arises from the requirement of parallel transport of vectors ( order - one tensors ) as well as for the requirement that higher order tensors lay down a scalar spacetime metric for invariant "lengths" that take the same value in every curvilinear coordinate system; whereupon, the 2nd - order metric tensor throughout the gravitational fabric of spacetime can be obtained from each spacetime point to some other "transported" spacetime point by parallel transport. In other words, find the metric in one region of spacetime and, hence, by parallel transport you will know the value or characteristic of an equivalent metric in any other region of spacetime by the of parallel transport! In effect, therefore, the Christoffel symbol mathematically represents the intensity of the composite gravitational force field under conditions of parallel ( gyroscopic ) transport, whereby an infinite number of gravitational potentials on the curvilinear spacetime fabric surface "arithmetize" the Christoffel!

For example, let's parallel transport contravariant vector components and so that

which becomes

or, more succinct mathematical shorthand notation,

Now replace the above ordinary partial derivative by a covariant derivative for parallel transport and then differentiating term by term, we get

Relativity Science Calculator Since we are parallel transporting vector components and , all components of these respective vectors in both direction and magnitude ( length ) remain constant ( think: gyroscopic motion ) and, hence, under differentiation these components all go to zero; therefore,

Referencing the earlier identity

set the covariant derivative of the 2nd - order metric tensor as follows:

Method 1: metric lowering indices:

Relativity Science Calculator Relativity Science Calculator Method 2: Using the original Christoffel definitions:

Relativity Science Calculator Ricci's Theorem Proof

Relativity Science Calculator § The Christoffel symbol in a pseudo - Euclidean space:

§ The Christoffel symbol in a General Relativity Riemannian space:

The non - tensor Christoffel symbols as "correction factors"

§ Transformation laws:

We define the transformation law for the covariant vector ( 1st - order tensor ) as

for a system of 4 - coordinates and , , transforming covariant of a covariant 4 - vector field in the "old" - system into the "new" - coordinate system of covariant components, we identify

as the cosine of the angle between the - axis of the "new" - coordinate system and the - axis of the "old" - coordinate system.

Relativity Science Calculator Hence, we can also equivalently write

§ Transforming the covariant ( tensor ) derivative of the covariant vector:

We already know that

Also as before, we define

Question: What is the transformation law for the partial derivative of the above covariant vector components?

Well, let's see:

But we also see that the partial derivative transformation

is "spoiled" from ever becoming a canonical tensor due to the second "hanging" term

However, by introducing a "correction factor" and cleaning up a bit, the above partial derivative transformation can be put back into a standard canonical form and hence "made to work" as follows:

Introducing

which defines the transformation law for the Christoffel symbols but which in turn precludes these "correction factors" as being tensors

Relativity Science Calculator due, no less again, to the "hanging" second term as a "spoiler". However, this mathematical construction is being employed owing to the above mathematics of parallel ( gyroscopic ) transport which shortly will be made still more obvious, subtle as it may be. That is,

Remember, also, parallel ( gyroscopic motion ) transport of freely moving test particles or bodies in gravitational fields is mathematically written as

Now, look at this pair of equations:

which is the canonical transformation law for the covariant ( tensor ) derivative of covariant vector components as we observe

In conclusion, therefore, all of the foregoing confirms the transformation law for 2nd - order covariant tensors since

thus confirming our earlier definition and construction for a 2nd - order tensor arising out of the covariant derivative of the covariant vector under conditions of parallel ( gyroscopic ) transport.

Relativity Science Calculator § Summary:

The Euler - Lagrange Equation of Body Motion

The Euler - Lagrange equation of motion is a differential equation for which the local maximum and minimum of a given function is optimally stationary - i.e., there is no further upper or lower limit in the final solution for a given differential function. This variational calculus was jointly developed by Leonhard Euler, Swiss mathematician, and Italian mathematician Joseph Lagrange in the 1750s.

The outstanding mathematical advantage of employing the Euler - Lagrange equation of motion for the Riemann geometry of curvilinear spacetime coordinates lies in its consistent uniformity between differing generalized coordinate systems.

Therefore, the final derivation for the Riemannian geodesic equation seeks the minimal world timeline transit path thru intervening gravitational spacetime for any object body or test particle, thereby expending the least amount of energy according to the general Principle of Least Action; hence, the Euler - Lagrange equation of motion which strives for a minimum becomes vitally central in the mathematical physics of general relativity!

Relativity Science Calculator Relativity Science Calculator The Derivation of the Geodesic Spacetime Equation for Bodies in Motion

From our earlier construction for the eventual derivation of the geodesic equation, we set the Lagrangian function for bodies in motion as

for the final geodesic world timeline we are attempting to mathematically develop, and for which the invariant spacetime interval

is the generalized Pythagorean Theorem in Riemannian spacetime geometry, "arithmetized" by the tensor metric . Therefore, we must solve the Euler - Lagrange equation of motion where

Continuing ...

Relativity Science Calculator Therefore, from the Euler - Lagrange equation

we get the geodesic spacetime equation for body motion as

However, from parallel transport using the calculus of variation, we know that

Nevertheless, how are equations and reconciled?

Relativity Science Calculator Relativity Science Calculator § Other representations of the geodesic equation under conditions of parallel ( gyroscopic ) transport thru gravity fields:

Relativity Science Calculator General Relativity Interpretation of

1). For flat Euclidean ( i.e., Newton's straightline inertial ) "Descartes motion" where in a straight - line, Euclidean - localized region contained exactly within a parallel transiting gyroscope along some geodesic world timeline, we get

2).

This interpretation is stating that for whatever force field is involved in parallel ( gyroscopic ) transport motion, both the 4 - velocity vector field as well as particle momentum are parallel ( gyroscopic ) transported along the test particle's geodetic world timeline.

3). Finally, the geodesics of curved Riemannian spacetime "fabric" geometry are exactly the world timelines of 4 - velocity vectors for freely moving test particles or other celestial bodies in motion according to the Principle of Least Action between separated spacetime points. More exactly, a geodetic world timeline is a Riemannian curve which parallel ( gyroscopically ) transports a tangent vector in a maximum amount of proper ( "wrist watch" ) time from a given spacetime point to another spacetime point, according to the Principle of Least Action.

Einstein's Diagram for Acceleration's Gravity Field

Einstein first creates an imaginary pure Galilean "world" in which there are no other masses of bodies or stars to effect anyone or anything, except that there does exist an observer floating freely in this pure in vacuo Galilean space. Einstein then places an imaginary "chest" enclosing this free floating observer which in turn serves as the observer's inertial frame of reference. However by some external "being", using Einstein's terminology, the enclosing chest for our free floating observer is converted to a non - inertial ( accelerating ) frame of reference by the accelerating pull or force made possible by this same external "being". Naturally our free floating observer will experience an accelerating field of gravity and the inherent quality of mass for the enclosed observer is made obvious in the following pictorial:

Relativity Science Calculator By giving a physical interpretation to Einstein's "Law of Equivalence of Acceleration and Gravity", Einstein also demonstrates the "Law of Equivalence of Inertial and Gravitational Mass" and, hence, does so within a powerfully new generalized postulate of relativity!!

[5] ∗ In all of Einstein's other writings, it is spacetime geometry itself which is pulling and pushing gravitational masses [ not mysterious "beings"! ] which in turn are deforming spacetime's malleable fabric; in this way, according to Einstein, gravitational masses are best understood in terms of non - Euclidian metric geometries for deformable spacetime fabrics and therefore General Relativity mathematics employs 4 - dimensional tensors, a specialized matrix algebra and calculus for curvilinear Riemannian surface geometries of which Euclid's cartesian geometry is a unique case. The Geodesic Equation

§ References:

1. "Vector and Tensor Analysis With Applications", by A.I. Borisenko and I.E. Tarapov, English translation from the original Russian by Richard A. Siverman, Dover Publications, Inc., New York

2. "Introduction to Vector and Tensor Analysis", by Robert C. Wrede, Dover Publications, Inc., New York

3. "Relativity: Modern Large - Scale Spacetime Structure of the Cosmos", by Moshe Carmeli, Editor ( born Baghdad, , 1933 - deceased Beer Sheva, , 2007 ), Albert Einstein Professor, Physics Department, Ben Gurion University, Beer Sheva, Israel, World Scientific Publishing Co., Pte Ltd.

4. "Gravitation and Spacetime", by Hans C. Ohanian and Remo Ruffini, W.W. Norton & Company, New York, London

5. "Gravitation and Inertia", by Ignazio Ciufolini and John Archibald Wheeler, Princeton Series in Physics

6. "Principles of Physical ", by P.J.E. Peebles, Princeton Series in Physics

7. "Tensors, Relativity, and Cosmology", by Mirjana Dalarsson and Nils Dalarsson, Elsevier Academic Press, Inc.

8. "Introduction to the Theory of Relativity, With a Forward by Albert Einstein", by Peter Gabriel Bergmann, Dover Publications, Inc., New York Relativity Science Calculator