Related Links (Outside this Site)

The Rigid Rotating Disk in Relativity by Michael Weiss.
Introduction to General Relativity by Gerard 't Hooft (Utrecht University)
Tensors and Relativity by Peter Dunsby (University of Cape Town, 1996).
General Relativity by David M. Harrison (University of Toronto).
Reflections on Relativity by Kevin S. Brown.
General Relativity Tutorial by John Baez.
About Black Holes... by Chris Hillman.
Einstein-Cartan theory by Heiko Herrmann et al. (2004).
Einstein-Cartan theory by Andrzej Trautman (arXiv, June 2006).
Cartan-Einstein Teleparallelism (TP) by José G. Vargas & Doug G. Torr.
On the History of Unified Field Theories by Hubert F.M. Goenner (2004).
Did Einstein cheat? by John Farrell (2000)

Wikipedia : Introduction to General Relativity | General Relativity | Einstein field equations
Cartan formalism | Spin tensor | Einstein-Cartan theory | (Lack of) Nordtvedt effect
Tipler Machine (time travel) | Nordström's metric theory of gravitation (1913, obsolete & falsified)
Variational methods in general relativity

The Mechanical Universe (28:46 each episode) David L. Goodstein (1985-86)

26 Beyond the Mechanical Universe (#27) 28

Introduction : In classical mechanics, it's often convenient to describe motion in non-inertial frames of reference (e.g., a rotating coordinate system). In such a system, the laws of mechanics won't hold unless we use particular expressions for the derivative of a vector (the acceleration and rotation vector of the frame of reference itself are involved). Alternately, we may apply to any frame of reference the laws of mechanics in the form they assume in an inertial frame, provided we introduce special fictitious forces proportional to the mass of the object (like the centrifugal force or the Coriolis force).

We could always bypass either approach and analyze the problem with respect to an inertial coordinate system (no matter how contrived a construction this inertial system may be). We did just that in our analysis of the Coriolis effect in free fall and the Sagnac effect.

Albert Einstein remarked that the force exerted by gravity on an object [which we call the weight of that object] is strictly proportional to its inertial mass, just like the aforementioned fictitious forces. He dubbed this observation the equivalence principle (i.e., inertial mass and gravitational mass are one and the same) and drew all the consequences of putting gravitational forces and inertial forces on the same footing.

The only difference between gravity and an ordinary fictitious force field is that the former cannot (usually) be reduced to a mere artifact of coordinate motion. So, with gravity, we no longer have the luxury of going back at will to an "inertial frame" where physical laws are simpler. Instead, we're stuck with a system of coordinates corresponding to whatever the local geometry becomes because of the presence of gravity. The ensuing mathematical framework is the stage for the General Theory of Relativity.

This stage was not left empty by Einstein, who came up with a compatible description of how gravity is produced by mass (or, rather, energy). This ends up relating the curvature of spacetime with the distribution of energy in it. The result is Einstein's field equations. The mathematics involved may be intimidating but the basic principles (stated above) are quite simple. The implications are mind-boggling.

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(2021年02月16日) Deflection of grazing starlight by the Sun
One famous early tests of General Relativity (Eddington, 1919).

According to (nonrelativistic) Newtonian mechanics, a corpuscle traveling at speed c which comes very near the surface of a spherical star (of mass M and radius R) gets deflected by the following small angle:

q = 2 M G / R c²

Einstein himself obtained this non-relativistic relation in 1911 using his principle of equivalence before he worked out the full machinery of General Relativity. Einstein's argument is presented in a lecture by Leonard Susskind. The above Noewtonian relation was obtained by Cavendish in 1784 and rediscovered in 1801 by Johann Georg von Soldner (1776-1833) who was the first to publish it (1804).

The above Newtonian relation is also easy to obtain rigorously directly from Kepler's laws. The area velocity of the corpuscule around the Sun is a constant C which can be otained in two different ways:

• At perihelion (when the light ray grazes the surface of the Sun, at a distance R from the center) the surface swept in time dt is a rectangular triangle of side R and c dt, whose area is ½ R c dt, So, C = ½ R c.
• ...

Eliminating C between those two equations yields the advertized relation:

q = 2 M G / R c²

The problem is that a Newtonian approximation can't be applied to something of zero mass. The deflection predicted by General Relativity is twice as large. It was first obtained by Einstein in 1916.

q = 4 M G / R c²

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Louis Vlemincq (2005年07月25日; e-mail) Observer on a Rotating Disc
Does the Harress-Sagnac effect contradict General Relativity ?

Mann muss immer generalizieren. Edward "Ned" van Vleck (1916)
(whose son, John Hasbrouck van Vleck, earned a Nobel prize in 1977)

The Sagnac effect is simply the observation that two beams of light circling the same rotating loop in opposite directions will take different times to go back to the starting point (simply because the starting point itself will have moved toward one beam and away from the other before light returns to it).

In the main, the version of the Sagnac effect which involves mirrors rather than fiber optics is nonrelativistic. In our introduction to the Sagnac effect, we've shown that special relativity implies that a Sagnac apparatus made from fiber optics works exactly like a mirrored one enclosing the same surface (regardless of the refractive index n of the optical cable used).

For some obscure reason, the Sagnac effect has been touted as "alternative science". It's not. In fact, the Sagnac effect has been providing a reliable solid-state substitute for gyroscopes aboard aircrafts for over 30 years.

A Sagnac apparatus normally rotates much too slowly to make general relativity quantitatively relevant. However, the study of the Sagnac effect is a great introduction to the concepts involved in general relativity (GR).

The example of the rotating disc is what convinced Einstein himself that Euclidean geometry was inadequate in a general coordinate system where an observer at rest would see masses accelerate from either of two equivalent causes: gravitational fields or nonuniform motion (with respect to a local Lorentzian "inertial" system).

[画像: Sagnac's Rotating Interferometer]

We've established elsewhere the following expression for the time lag in the respective returns of two light beams traveling in oppositite directions around a circular loop of radius R, rotating around its axis at a rate w.

Dt = 4p R ² w

vinculum

c ² - w ² R ²

This expression is valid for an inertial [nonrotating] observer who does not move with respect to the loop's center of rotation. The main reason for the observed nonzero lag time Dt is that each beam must travel a different distance to reach the half-silvered mirror which moves with the loop. A careful analysis with fiber optics reveals that Dt does not depend on the index of refraction (n) and is the same for a mirrored apparatus as well (n = 1).

It's enlightening to ponder the above expression, which we may rewrite:

Dt = ( 4 / c² ) W . S

vinculum

1 - (W´r)² / c²

Note the bold type indicating vectorial quantities, namely:

• r is the 3D position, relative to an origin on the axis of rotation.
• W is the axial rotation vector (cf. usual sign convention). || W || = w
• S is the loop's vectorial surface, an axial vector which depends not only on the conventional orientation of space but also on which direction is chosen as positive to travel around the loop. In Euclidean geometry, S may be defined by a contour integral around the oriented loop (C+).

S = ½ ò_C+ r ´ dr

For a closed loop C+ this defining integral does not depend on the arbitrary origin chosen for the position vector r. Anybody encountering this for the first time is encouraged to work out S explicitly for a circle of radius R, with the following parametric equations (0 < q < 2p).

x = a + R cos q ; y = b + R sin q ; z = c

Now, the denominator in the above looks like a relativistic correction (indeed it is) which we may discard at first

[画像: Come back later, we're still working on this one... ]

Sagnac Time Lag (observer tied to the loop)

Dt' = 4 W . S

vinculum

c ²

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(2005年07月29日) Solid in Relativistic Motion
A rigid motion is a state of equilibrium, which can change only so fast.

In classical mechanics, a solid is a body whose parts always remain at the same distances from each other, in what's called rigid motion. In such a motion there must be a rotation vector W which ties the velocities of any pair A and B of the solid's points, via the following relation ( W is an axial vector whose sign depends on space "orientation").

v_A - W ´ A = v_B - W ´ B

This is only a good approximation to physical reality if any change in the velocity of a point is somehow made known instantly throughout the solid so that the relative distance of all pairs of its points can be maintained...

In practice, however, such information can be propagated no faster than the speed of sound within the solid. Loosely speaking, a change in rotation which starts at the axis of rotation will propagate at the slower transverse speed, while other changes propagate at a speed intermediary between this speed (S-waves) and the true speed of sound (P-waves).

In classical mechanics, the assumption is made that the damped vibrations which enforce "solid" motion are fast enough (and small enough) to be neglected.

This is true in relativistic mechanics also, but only if changes in speed and rotation are slow enough compared to what changes them (namely sound). This usually makes a relativistic treatment virtually useless, except in the stationary cases: a "solid" may have been put in rapid rotation quite violently, but its ultimate state is an unchanging state of equilibrium which may be worth studying. (Even so, it's fallacious to consider a solid with parts moving faster than light !)

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(2009年07月25日) Contravariance and Covariance in the Euclidean Plane
A gentle introduction (m = 1,2) to tensor notations (m = 0,1,2,3).

The Euclidean plane is a two-dimensional vector space endowed with a norm (i.e., length of a vector) induced by a definite positive dot product (whereby the "square" of any nonzero vector is positive). In this familiar context of classical geometry, relativistic tensor notations and concepts can be nicely illustrated on paper (using compass and straightedge, if need be).

Warning : A symbol with an index appearing as a superscript is different from the same symbol with the same index as a subscript !

We consider a basis of two linearly independent vectors, ê₁ and ê₂ which need not be orthogonal and need not be of unit length...

Any infinitesimal two-dimensional vector dV is a linear combination whose coefficients are said to be its [contravariant] coordinates in that basis:

dV = dv¹ ê₁ + dv² ê₂

By definition, the covariant coordinates of dV (endowed with lower indices) are obtained by dotting dV into ê₁ and ê₂ respectively:

dV . ê₁ = dv₁ and dV . ê₂ = dv₂

For an orthonormal basis, those are equal to the contravariant coordinates. Otherwise, they are regular coordinates (i.e., coefficients in a linear combination) for the dual basis consisting of two other vectors, ê¹ and ê² defined by:

ê₁ . ê¹ = 1 ê₁ . ê² = ê₂ . ê¹ = 0 ê₂ . ê² = 1

Those defining relations can be summarized using the Kronecker delta symbol, in a way which remains true in any number of dimensions:

ê _i . ê ^j = d_i^j ( equal to 1 if i = j and zero otherwise)

dV = dv₁ ê¹ + dv₂ ê² = (dV.ê₁ ) ê¹ + (dV.ê₂ ) ê²

= dv¹ ê₁ + dv² ê₂ = (dV.ê¹ ) ê₁ + (dV.ê² ) ê₂

|| dV || ² = dv¹ dv₁ + dv² dv₂

All this is best expressed by introducing the metric tensor g_mn (which defines the dot product and, hence, the notion of length). g_mn is symmetric ( g₁₂ = g₂₁ ) because the dot product is commutative ( ê₁ . ê₂ = ê₂ . ê₁ ).

ê₁ = g₁₁ ê¹ + g₁₂ ê² = (ê₁ . ê₁ ) ê¹ + (ê₁ . ê₂ ) ê²

ê₂ = g₂₁ ê¹ + g₂₂ ê² = (ê₂ . ê₁ ) ê¹ + (ê₂ . ê₂ ) ê²

By definition, the square of the length of dV is the dot product dV . dV :

|| dV || ² = ( dv¹ ê₁ + dv² ê₂ ) . ( dv¹ ê₁ + dv² ê₂ )

= g₁₁ dv¹ dv¹ + ( g₁₂ + g₂₁ ) dv¹ dv² + g₂₂ dv² dv²

= dv¹ dv₁ + dv² dv₂

= g¹¹ dv₁ dv₁ + ( g¹² + g²¹ ) dv₁ dv₂ + g²² dv₂ dv₂

The matrix g^mn that appears in this last expression is the multiplicative inverse of g_mn which expresses the reciprocal linear relations, namely:

ê¹ = g¹¹ ê₁ + g¹² ê₂ = (ê¹ . ê¹ ) ê₁ + (ê¹ . ê² ) ê₂

ê² = g²¹ ê₁ + g²² ê₂ = (ê² . ê¹ ) ê₁ + (ê² . ê² ) ê₂

The following relation holds in any number of dimensions:

å _k g _ik g ^kj = d_i^j (Kronecker's delta symbol)

[画像: A unit circle that looks like an ellipse... ] Now, the "square" grid is a luxury that's not needed. The metric properties of the original basis (black vectors) are entirely specified by the metric tensor, which gives the shape of the unit circle (orange) as a specific cartesian equation in that basis. The reciprocal basis vectors (red) are also specific linear combinations of the original vectors which depend only on the metric tensor...

Summary :

In a metric space, there's only one kind of vector, which may be specified either by its contravariant coordinates or its covariant coordinates :

V = V¹ ê₁ + V² ê₂ = V₁ ê¹ + V₂ ê²

The distinction between "contravectors" and "covectors" is misguided because the metric itself establishes a firm one-to-one correspondence between them. (Such a distinction is only useful for a general vector space not endowed with a metric.)

The metric tensor and its inverse can be used to switch back and forth between the contravariant and the covariant representations of any vector:

V_i = å _j g _ij V ^j and V ⁱ = å _j g ^ij V_j

The backdrop of general relativity is essentially a generalization of this to four dimensions with a metric of signature - + + + (as opposed to + + for the Euclidean plane discussed above) which singles out the particular dimension of time. Technically speaking, relativistic spacetime is a Lorentzian manifold, a particular case of semi-Riemannian manifold.

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(2009年08月07日) Contravariance and Covariance in the Lorentzian Plane
Introducing a Lorentzian metric in the plane.

Let's do over the previous numerical example with a Lorentzian dot product :

( x, t ) . ( x', t' ) = x x' - t t'

[画像: Come back later, we're still working on this one... ]

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(2009年08月05日) Tensors in metric spaces
What tensors really are.

By definition, the scalars of a vector space are its tensors of rank 0.

In any vector space, a linear function which sends a vector to a scalar may be called a covector. Normally, covectors and vectors are different types of things. (Think of the bras and kets of quantum mechanics.) However, if we are considering only finitely many dimensions, then the space of vectors and the space of covectors have the same number of dimensions and can therefore be put in a linear one-to-one correspondence with each other.

Such a bijective correspondence is called a metric and is fully specified by a nondegenerate quadratic form, denoted by a dot-product ("nondegenerate" precisely means that the associated correspondence is bijective).

A metric is said to be Euclidean if it is "positive definite", which is to say that V.V is positive for any nonzero vector V. Euclidean metrics are nondegenerate but other metrics exist which are nondegenerate in the above sense without being "definite" (which is to say that V.V can be zero even when V is nonzero). Such metrics are perfectly acceptable. They include the so-called Lorentzian metric of four-dimensional spacetime, which is our primary concern here.

Once a metric is defined, we are allowed to blur completely the distinction between vectors and covectors as they are now in canonical one-to-one correspondence. We shall simply call them here's only one such type, now). A tensor of rank zero is a scalar.

More generally, a tensor of nonzero rank n (also called n^th-rank tensor, or n-tensor) is a linear function that maps a vector to a tensor of rank n-1.

Such an object is intrinsically defined, although it can be specified by either its covariant or its contravariant coordinates in a given basis (cf. 2D example).

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(2009年07月29日) Signature of a Quadratic Form
Bases in which a given metric tensor has its simplest expression.

In the previous introductory article, we defined the metric tensor with respect to a particular basis in terms of a known ordinary euclidean dot product:

g _ij = ê _i . ê _j

From that metric tensor alone, we computed reciprocal vectors satisfying:

ê _i . ê ^j = d_i ^j

It turns out that this can always be done if we define ab initio our "dot product" (which need not result in a positive definite quadratic form) by specifying the aforementioned metric tensor to be any given symmetric matrix (invertible or not). Furthermore, there are special vector bases where the dot product so defined has a particularly simple expression, namely:

• g _ij = 0 when i differs from j.
• g _ii is equal to 0, -1, or +1.

More loosely, we only need a matrix M such that M g M* is diagonal. In all such cases, the numbers of negative and positive quantities on the diagonal are the same and they define what's called the signature of the metric. (If there are no zeroes on the diagonal, the metric is said to be nondegenerate.)

Rene Descartes 1596-1650 One easy way to determine the signature of a given metric tensor (or any hermitian matrix, actually) is to use Descartes' rule of signs (1637) on its characteristic polynomial (whose roots are all real).

[画像: Come back later, we're still working on this one... ]

ê _i . ê ^j = g _i^j = d _i^j always (by definition of the reciprocal vectors).

ê _i . ê_j = g_ij = h _ij in an orthonormal basis only.

For a nondegenerate metric, h_ij = 0 when i ¹ j whereas h_ii = ± 1.

Metric Signature | Sign conventions | Sylvester's law of inertia (1852)

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(2009年07月21日) Covariant and Contravariant Coordinates
Displacements are contravariant, gradients are covariant.

In the context of general relativity, a point M in spacetime (also called an event ) is determined by 4 real numbers, called coordinates denoted by superscripted variables in one "coordinate system" or the other:

M = ( x⁰, x¹, x², x³ ) _[x] = ( y⁰, y¹, y², y³ ) _[y]

Displacements and other contravariant coordinates :

The value of each coordinate y ^m is a function of the event itself and is, therefore, a function of all four x-coordinates. The differential of each y-coordinate is thus a linear combination of the differentials of the four x-coordinates. By definition, the coefficients of those linear combinations are known as partial derivatives :

d y ^m = ¶y^m dx⁰ + ¶y^m dx¹ + ¶y^m dx² + ¶y^m dx³

vinculum vinculum vinculum vinculum

¶x⁰ ¶x¹ ¶x² ¶x³

The partial derivative with respect to one named spacetime coordinate is understood to be the derivative obtained by holding constant the other coordinates by the same name (carrying a different index).
Without such a convention, a partial derivative would lose its meaning as soon as it becomes isolated from a well-defined "total differential" formula (of which the above sum is a typical example).
No such uniform convention is possible in thermodynamics, where the "other" variables which are held constant must routinely be given as subscripts to a pair of parentheses surrounding the curly expression.

In relativistic tensor calculus, such sums are rarely written out explicitly. Instead, the Einstein summation convention is used, which states that a multiplicative expression where an index occurs twice (once downstairs and once upstairs) denotes the sum of 4 terms where that index takes on all values from 0 to 3. Thus, the above sum is equivalent to:

d y ^m = ¶y^m d x ⁿ

vinculum

¶xⁿ

The Einstein summation convention applies recursively: Thus, an expression with two pairs of repeated indices would stand for a sum of 16 terms, three such pairs would denote a sum of 64 terms, etc.

If the four coordinates of a vectorial quantity V obey the transformation rules that we just established for an infinitesimal spacetime displacement, they are called contravariant coordinates and bear superscripted indices:

V ^m [y] = ¶y^m V ⁿ [x]

vinculum

¶xⁿ

Instead of using different sets of names, we may underscore whatever relates to the second frame of reference (vectorial components, coordinates, differential operators with respect to coordinates, etc.). The above becomes:

Transformation of Contravariant Coordinates

V^m = ¶_n x^m Vⁿ

Each vector ê of a local reference frame is identified with a lower index from 0 to 3, to conform to the standard restriction, which says that a summation index must appear once as a subscript and once as a superscript:

Expansion of a vector using contravariant coordinates

V = V ^m ê _m = V⁰ ê₀ + V¹ ê₁ + V² ê₂ + V³ ê₃

The quantity on the left-hand side lacks any "open" index because we are referring to the mathematical object itself, as opposed to its coordinates in a particular frame of reference. We shall henceforth use bold type to denote an object with components that are not made explicit by an apparent index (loosely speaking, there are hidden indices in a bold symbol).

There's also an implication that a given object could be described by other schemes besides the aforementioned contravariant linear combinations. Indeed, one such scheme is the covariant viewpoint which we are about to describe (both aspects become interchangeable in a metric space).

Gradients and other covariant coordinates :

Transformation of Covariant Coordinates

V_m = ¶_n x^m V_n

[画像: Come back later, we're still working on this one... ]

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(2009年07月21日) The metric tensor g_mn and its inverse g^mn
Lowering or raising indices.

The spacetime interval (squared) is g_mn dx^m dxⁿ

[画像: Come back later, we're still working on this one... ]

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(2009年07月25日) Duality
A dual is obtained by switching all indices (and complex conjugation).

(V_m )^* = (V^*) ^m (V ^m )^* = (V^*) _m

[画像: Come back later, we're still working on this one... ]

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(2009年07月23日) Lower and upper partial derivatives
Derivatives with respect to contravariant or to covariant coordinates.

Loosely speaking, a lower index at the denominator becomes an upper index for the overall ratio, and vice-versa.

Thus, the derivative with respect to a contravariant coordinate carries a lower index whereas the derivative with respect to a covariant coordinate carries an upper index. Those two operators applied to y are respectively denoted:

y_,m = ¶_my = ¶y

vinculum

¶x^m

y^,m = ¶ ^m y = ¶y

vinculum

¶x_m

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G s
nm

(2009年07月30日) Christoffel Symbols
Coordinates of the partial derivatives of the basis vectors.

The basis we choose for local vectors and tensors may vary [smoothly] from one spacetime point to the next. That variation must be accounted for.

Likewise, in Newtonian mechanics, a "moving" coordinate system entails the introduction of a rotation vector and of various forces proportional to mass (inertial, centrifugal, Coriolis, Euler).

The so-called coefficients of affine connection are simply the coordinates of the partial derivatives of the basis vectors. They are better known as Christoffel symbols (or gammas) and may be defined as follows:

G s
nm = ê ^s . ¶_mê _n

Since ê ^s . ê _n = d^s_n is a constant, its derivatives vanish and we have:

- G s
nm = ê _n . ¶_mê ^s

It's best to maintain the order of the downstairs indices (the differentiation index (m) should be placed last ) although that order is irrelevant in Einstein's [standard] General Relativity because of the symmetry induced by the equivalence principle. A few authoritative references support that convention:

• Misner et al. (1973) Equation 8.19a, page 209. [Strongly!]
• Schutz (1985) Equation 5.43, page 135.

Some reputable authors (including Weinberg and Wald) shun the above asymmetrical definition and/or invoke immediately the symmetry induced by the equivalence principle. When discussing standard General Relativity, many authors don't even bother with a consistent order of the Christoffel indices.

Without the symmetry of the [connection] coefficients,
we obtain the twisted spaces of Cartan [1922],
which have scarcely been used in physics so far,
but which seem destined to an important role.
Léon Brillouin 1938

Wikipedia : Christoffel symbols | Elwin Bruno Christoffel (1829-1900)

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Ñ = ê ^m Ñ_m

(2009年07月29日) Absolute nabla operator Ñ
Introducing the covariant derivatives Ñ_m

Covariant derivatives are due to Gregorio Ricci-Curbastro (1853-1925) who invented most of tensor calculus between 1884 and 1894 (Delle derivazione covariante e contravariante, Padova, 1888). In 1900, with his former student Tullio Levi-Civita (1873-1941) Ricci published a 75-page masterpiece entitled Méthodes de calcul différentiel absolu et leurs applications. That treatise unified and extended the pioneering efforts of Carl Friedrich Gauss (1777-1855), Bernhard Riemann (1826-1866) and Elwin Christoffel (1829-1900).

It was Marcel Grossmann (1878-1936) who brought that work to the attention of Albert Einstein when Einstein asked him for help in formulating a relativistic theory of gravitation (Grossmann and Einstein had been classmates at ETH Zürich). At the time, Newtonian gravity was known to be incompatible with Special Relativity and Paul Ehrenfest (1880-1933) had pointed out (in 1909) the noneuclidean character of geometry in one particular noninertial frame of reference: The rim of a spinning circular platform measures less than p times its diameter ! Equating inertial accelerations and gravitational fields (his principle of equivalence) Einstein suspected that gravity might be related to a local disturbance in the metric features of spacetime...

A tensor field is a function (linear or not) mapping a spacetime point m to some tensor T of rank n. The linear function mapping an infinitesimal (vectorial) displacement dm to the corresponding variation of T is thus a tensor of rank n+1 which is denoted Ñ T . By definition:

Ñ T ( dm ) = d [ T ( m ) ]

Loosely speaking, that's also equal to T ( m + dm ) - T ( m )

The covariant derivative Ñ_m is to the absolute differentiation of a tensor T what the partial derivative ¶_m is to the differentiation of a scalar f.

Ñ T = ê ^m Ñ_m T d f = dx^m ¶_m f

Formally, Ñ_m can thus be defined by dotting Ñ into ê_m

Ñ_m = ê_m . Ñ

The crucial difficulty is that a tensor of nonzero rank is obtained by summing every coordinate multiplied into a matching tensor-product of basis vectors. The derivative of every such term is obtained by the product rule. Only one component is obtained by differentiating the coordinate itself; all the other components involve derivatives of the basis vectors. We'll first restate this in the case of tensors of rank 1 (vectors) before generalizing again to tensors of any rank.

A vector is really a linear combination of basis vectors which may well change as the differentiation variable varies. Therefore, the product rule fully applies (that's similar to the way rigid motion brings about a rotation vector ) and we obtain:

Ñ_m( V ) º ¶_m( V_n êⁿ ) = ¶V_n êⁿ + V_n ¶_m êⁿ

vinculum

¶x^m

So, ê ^m ¶_m( V_n êⁿ ) = ¶V_n ê ^m êⁿ + (ê ^m ¶_m êⁿ ) V_n

vinculum

¶x^m

= ¶V_n ê ^m êⁿ + (ê ^m ¶_m ê^s ) V_s

vinculum

¶x^m

Since ¶_m and Ñ_m have the same effect on basis vectors, what appears in the last bracket is actually the nabla operator Ñ = ê ^m Ñ_m applied to ê ^s. The coordinates of that are the Christoffel symbols introduced above:

Ñê ^s = - G s
nm ê ^m ê ⁿ

¶_mê ^s = - G s
nm ê ⁿ

The latter equation implies the former, which we plug into the above to obtain the following expression for the coordinates of the covariant derivative of a vector:

Covariant derivative of a vector

Ñ_m V_n º V_n;m = ¶V_n - G s
nm V_s

vinculum

¶x^m

The covariant derivative of a tensor of rank n entails a sum of n+1 terms:

T _ab;m = ¶T_ab - G s
ma T_sb - G s
mb T_as

vinculum

¶x^m

U _abg;m = ¶U_abg - G s
ma U_sbg - G s
mb U_asg - G s
mg U_abs

vinculum

¶x^m

If upper indices are used, the coordinates of contravariant derivatives obey a similar rule with the same symbols, but different summations and opposite signs:

Ñ_m V ⁿ º V ⁿ_;m = ¶V ⁿ + G n
sm V ^s

vinculum

¶x^m

Here's one example with mixed indices (one upstairs, two downstairs):

U _a^b_g;m = ¶U_a^b_g - G s
am U_s^b_g + G b
sm U_a^s_g - G s
gm U_a^b_s

vinculum

¶x^m

This simple statement summarizes 256 formulas, with 13 terms each...

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Ñ = ê_mÑ ^m

(2009年08月03日) Contravariant derivatives Ñ ^m
Rare differentiation along covariant coordinates.

The tensorial operator Ñ obeys the standard rules for raising and lowering of indices. This is consistent with its two equivalent (dual) expressions:

Ñ = ê_mÑ ^m = ê ^m Ñ_m

Indeed, by dotting everything into êⁿ we obtain:

êⁿ . Ñ = Ñ ⁿ = g^mn Ñ_m

That was merely a consistency check: Since the covariant derivative of a tensor is known to be a tensor, we are certainly allowed to raise the index which appears downstairs after a covariant differentiation...

We may also obtain expressions for contravariant derivatives ab initio :

Ñ ^m( V ) º ¶ ^m( V_n êⁿ ) = ¶V_n êⁿ + V_n ¶ ^m êⁿ

vinculum

¶x_m

So, ê_m ¶ ^m ( V_n êⁿ ) = ¶V_n ê_m êⁿ + (ê_m ¶ ^m êⁿ ) V_n

vinculum

¶x_m

= ¶V_n ê_m êⁿ + ( Ñ ê^s ) V_s

vinculum

¶x_m

The last term is just what we have in the same circumstances for covariant derivatives but we must now express it over a different tensorial basis (matching that of the first term) to obtain proper component-wise relations:

Ñê ^s = - G s
nm ê ^m ê ⁿ = - G s
nl g ^lm ê_m ê ⁿ

We may thus introduce the following not-so-common notation to simplify the explicit expressions for contravariant derivatives given below:

G sm
n = G s
nl g ^lm

Contravariant derivatives of a vector

Ñ ^m V_n º V_n^;m = ¶V_n - G sm
n V_s

vinculum

¶x_m

Ñ ^m V ⁿ º V ^n;m = ¶V ⁿ + G nm
s V ^s

vinculum

¶x_m

This is just for completeness... Those explicit formulas for contravariant derivatives are rarely used, if ever. They can be generalized to tensors of higher ranks by using the same patterns as covariant derivatives.

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(2009年10月21日) Variance of Christoffel Symbols & Cartan Tensor
The antisymmetric part of Christoffel symbols form a tensor.

The Christoffel symbols do not form a proper tensor (if they did, the above formulas for covariant derivation could be used to prove that the ordinary derivatives of a tensor form a tensor, which is not the case in curved space).

As shown below, the Christoffel symbols in two distinct frames (K and K) are related by equations involving both the the first and second derivatives of one set of coordinates with respect to the other set. It is the presence of second derivatives which indicates that Christoffel symbols are not tensors. However, the symmetry of those second derivatives make them vanish from the transformation rule for the asymmetric part of the Christoffel symbols. Those do transform like a proper tensor; they form a tensor, the Cartan tensor Q, which describes what's called the torsion of spacetime.

Q ^s_mn = ½ ( G s
mn - G s
nm )

Elie Cartan first described spaces with nonzero torsion in 1922 (Einstein's equivalence principle implies Q = 0 but this is not a logical requirement). When zero torsion is not assumed, General Relativity becomes what's known as Einstein-Cartan theory (Einstein himself thought that Q might describe electromagnetism; it doesn't). Spacetime torsion is needed if intrinsic pointlike spin is allowed.

Formally,

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(2009年10月15日) Einstein's Equivalence Principle (1907)
A postulate implying the symmetry of Christoffel symbols.

The Christoffel symbols do not form a tensor, but the following quantity is a proper antisymmetric tensor called Cartan torsion (or Cartan tensor ) :

Q ^s_mn = ½ ( G s
mn - G s
nm )

As argued below, Einstein's equivalence principle demands a zero torsion Q. However, that's not an absolute requirement: "gravity with torsion" will make intrinsic spin an asymmetric source of gravitation (besides the ordinary symmetrical stress tensor). This viewpoint is the basis for the Einstein-Cartan gravity (1922) advocated by Elie Cartan, Dennis Sciama, Tom Kibble, Richard J. Petti, etc.

The principle of equivalence postulated in Einstein's general theory of relativity implies that spacetime is torsion-free as it demands that there's always a local frame of reference (in "free fall") which is locally inertial.

Indeed, in a local inertial frame of reference, all the Chrisfoffel symbols vanish and, therefore, the torsion vanishes. Since it is a tensor, torsion must vanish in any other frame of reference as well, which means that Christoffel symbols are always symmetrical with respect to their two lower indices.

If such a torsion-free spacetime is metric-compatible, then the Christoffel symbols are functions of the metric coefficients and their first derivatives:

Torsion-free Christoffel symbols :

G s
mn = ½ g ^sl ( ¶_m g _ln + ¶_n g _ml - ¶_l g _mn )

Proof : Although the Christoffel symbols do not form a proper tensor, we may still introduce the following notation which will clarify the discussion:

G_smn = g_sr G r
mn

This turns metric compatibility into any of the following 3 equivalent equations:

¶_m g _ln = G _lnm + G _nlm

¶_n g _ml = G _mln + G _lmn

- ¶_l g _mn = - G _mnl - G _nml

In the torsion-free case, we may use the symmetry of G with respect to its last two indices and the addition of those three equations yields:

¶_m g _ln + ¶_n g _ml - ¶_l g _mn = 2 G _lnm

The advertised result is obtained by multiplying both sides into ½ g ^sl QED

Note that, conversely, the above formula only holds in the torsion-free case (as it does give Christoffel symbols that are symmetrical with respect to their last two indices). It also implies the metric compatibility which was used to derive it (the reader may want to check algebraically that the covariant derivatives of the metric tensor vanish when the Christoffel symbols have those advertised values).

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(2009年10月23日) Spacetime torsion with 4 independent components
What if the torsion Q_abg is a totally antisymmetric tensor...

With nonzero torsion in a metric-compatible geometry, the final summation in the above proof yields the following equation:

¶_m g _ln + ¶_n g _ml - ¶_l g _mn = G _lnm + G _lmn + 2 [ Q _nlm + Q _mln ]

It is tempting to consider the case where the square bracket vanishes. Since Q is already known to be antisymmetric with respect to its last two indices, this additional antisymmetry would make it a totally antisymmetric tensor.

In that case, the formula of the previous section just gives the symmetric part of the Christoffel symbols and, therefore, its generalization becomes:

G s
mn = ½ g ^sl ( ¶_m g _ln + ¶_n g _ml - ¶_l g _mn ) + Q ^s_mn

Conversely, such connection coefficients involving a totally antisymmetric torsion Q describe a metric compatible affine geometry.

This is so because we may split the covariant derivative of the metric tensor into a symmetric and an antisymmetric part. The former vanishes for the same algebraic reasons that make it vanish in the torsion-free case. The remaining antisymmetric part boils down to:

g _ij;m = - Q^s_im g _sj - Q^s_jm g _is = - Q_jim - Q_ijm = 0

In 4 dimensions, a completely antisymetric tensor of rank 3 has C(4,3) = 4 independent components. It may be obtained by applying to some vector the (essentially unique) totally antisymmetric tensor of rank 4. In other words, this kind of torsion can be described by a vector field...

Torsion Gravity by Richard T. Hammond (2002)
The Einstein-Cartan Theory by Andrzej Trautman (2005)
Wikipedia : Einstein-Cartan theory

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(2009年08月15日) Levi-Civita symbols: e_ij , e_ijk , e_ijkl , e_ijklm , etc.
Antisymmetric with respect to any pair of indices.

What I like best about Italy: Spaghetti and Levi-Civita.
Albert Einstein (1879-1955)

In dimension n, a totally antisymmetric tensor of rank k depends on C(n,k) independent components. When n = k, all such tensors are proportional.

Hodge duality / Jacobian of coordinate transforms... In dimension n, totally antisymmetric tensor of rank k is also called a k-vector. Hodge duality is a linear bijection between k-vectors and (n-k)-vectors. W. V. D. Hodge (1903-1975).

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Wikipedia : Levi-Civita Symbols | Tullio Levi-Civita (1873-1941)

Metric Compatibility

g _ab;m = 0

This is virtually an axiom nowadays (like the Pythagorean theorem has become an axiom defining distance in modern Cartesian geometry). Metric compatibility demands that the dot product of two parallel-transported vectors remain constant.

The situation is simpler than it sounds. One elementary way to visualize it is to consider the special case of a two-dimensional curved surface in Euclidean three-dimensional space... If the quadratic form corresponding to the metric tensor on that surface actually describes the 3D Euclidean metric, then it follows that it's invariant in the absolute sense underlying covariant differentiation.

The same would hold true for a curved "surface" of any dimension embedded in any "straight" space of higher dimension (endowed with a coordinate system where the higher-dimensional metric tensor is constant).

Once this remark is made, the expression of the Christoffel symbols in term of the metric coefficients can be obtained and we can forget about the crutch (or luxury) of being able to reason in a higher-dimensional space with a simpler structure.

Although that simpler encompassing structure may not exist, the relation between Christoffel symbols and metric coefficients which is derived from that mere possibility is given the name of metric compatibility.

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In a freely falling cartesian frame of reference, the components of the metric tensor are constant and the Christoffel symbols vanish. Thus, the covariant derivatives of the metric tensor vanish in this frame of reference and, therefore, in any other.

Some trivial consequences of metric compatibility

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(2009年07月31日) The Curvature Tensors
The Ricci tensor is a contraction of the Riemann curvature tensor.

The Riemann tensor (also called Riemann-Christoffel tensor ) is a tensor of rank 4 related to the the commutator of covariant derivatives as follows:

[ Ñ_m , Ñ_n ] V ^a = Ñ_m (Ñ_nV ^a ) - Ñ_n (Ñ_mV ^a )

= R ^a_bmn V ^b

Expressing covariant derivatives in terms of Christoffel symbols, we obtain:

The Riemann-Christoffel curvature tensor :

R ^a_bmn = ( ¶_m G a
bn + G a
ml G l
bn ) - ( ¶_nG a
bm + G a
nl G l
bm )

All the symmetries of the Riemann curvature tensor are best expressed after putting all its indices downstairs (by lowering the first index in the above):

R _abcd = - R _bacd = - R _abdc = R _cdab

R _abcd + R _adcb + R _acdb = 0

Thus, the 256 elements depend "only" on 20 independent components.

The Ricci tensor is a symmetrical tensor of rank 2 obtained by a contraction of the Riemann tensor. Both tensors can be denoted by the same symbol (R) because there's (usually) no risk of confusion, as they have different ranks :

The Ricci curvature tensor :

R _mn = R ^l_mln

Because of the symmetries of the Riemann tensor, the Ricci tensor is (up to a sign change) the only nonvanishing contraction of the Riemann tensor.

Wikipedia : Ricci curvature tensor | Riemann-Christoffel curvature tensor
Curvature of Riemannian manifolds

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(2009年08月08日) Bianchi Identity & Einstein Tensor

Luigi Bianchi (1856-1928) rediscovered the identities named after him in 1902. They had first been discovered in the early 1880's by his former classmate Gregorio Ricci-Curbastro who had forgotten all about it (according to Tullio Levi-Civita, the main collaborator and only former doctoral student of Ricci's).

The Bianchi identity:

R _abmn;l + R _ablm;n + R _abnl;m = 0

A contracted version holds for the Ricci tensor (HINT: multiply by g ^am ).

Contracted Bianchi identity:

R _bn;l - R _bl;n + R ^m_bnl;m = 0

By contracting this with respect to the indices b and n, we obtain:

R ⁿ_n;l - R ⁿ_l;n + R ^mn_nl;m = 0

The Ricci scalar R = R ⁿ_n appears in the first term. The second and third terms happen to be equal. So, the whole relation boils down to:

R _;l - 2 R ^m_l;m = 0

That key relation establishes that the following tensor, introduced by Einstein, has a vanishing divergence (i.e., G^m_n;m = 0 ).

Definition of Einstein's Tensor:

G_mn = R_mn - ½ g_mn R

Besides the metric tensor g itself, the Einstein tensor G turns out to be the only divergence-free second-rank tensor that can be built from the Riemann curvature coefficients.

That simple remark (which is not so easy to prove) makes the forthcoming Einstein field equation look almost unavoidable as a mere linear dependence (involving two fundamental constants of nature, L and G) between the three prominent divergenceless second-rank tensors g, G and T. The third of those is the stress tensor T, discussed next, whose lack of divergence expresses the conservation of energy and momentum.

Bianchi identities | Einstein tensor

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(2020年09月17日) The Weyl Tensor
Traceless component of the Riemann tensor.

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Weyl tensor | Hermann Weyl (1885-1955)

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(2009年07月31日) The Stress Tensor (i.e., energy density tensor )
Flow of energy density is density of conserved linear momentum.

As a conserved quantity, energy has a flow vector which is linear momentum. Together, energy and momentum form a quadrivector whose components are all conserved quantities. The 4-dimensional flow of that quadrivector is a tensor of rank 2 whose spatial components have the dimension of a pressure; it's called the stress tensor. (Also called stress-energy or energy-momentum-stress.)

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(2005年08月22日) Einstein's Field Equations (Einstein, 1915年11月11日)
Presented to the Prussian Academy of Science on November 25, 1915.

Matter tells space how to curve,
and space tells matter how to move.
John Archibald Wheeler (1911-2008)

Einstein's Law of the Gravitational Field

( R_mn - ½ g_mn R ) + L g_mn = 8 p G T_mn

vinculum

c ⁴

The symbols in this relation have the following meanings:

• g_mn is the metric tensor (describing the gravitational potential).
• R_mn is the second-rank curvature tensor (the Ricci tensor ).
• R = g^mn R_mn = R^m_m is the scalar curvature (or Ricci's scalar ).
• T_mn is the stress-energy tensor (pressure = density of energy).
• G is Newton's constant of gravity (about 6.67428(67) ´ 10^-11 SI ).
• L is the cosmological constant (only introduced in 1917).
• R_mn - ½ g_mn R = G_mn is Einstein's tensor.

The elements of the stress tensor T are in units of energy density or pressure (same thing; a pascal is a joule per cubic meter or a newton per square meter).

If the coordinates are all in distance units (they need not be) then the metric tensor is dimensionless and the intrinsic curvatures are homogeneous to the reciprocal of a surface area. So is the cosmological constant.

Einstein introduced it in 1917 for the wrong reasons, as he remarked that such a cosmological term would allows the Universe to be static (neither expanding nor contracting). That justification was misguided for two reasons: First, Edwin Hubble (1889-1953) would establish a few years later (1929) that the Universe is not static (it's expanding at a rate of about 70 km/s/Mpc). Second, a Universe obeying Einstein's complete equation would not durably remain in equilibrium (the slightest perturbation in the distribution of its contents would get amplified and result in a nonstatic universe). A zero cosmological constant was compatible with observations made during the lifetime of Einstein. That's why he once said that introducing it was "the worst mistake of [his] life".

With hindsight, the discussion should have focused on the actual value of the cosmological constant, since the mathematics allow anything. Surprisingly, this constant is neither zero (as Einstein thought in 1929) nor negative (as Einstein had thought in 1917).

We now know that the cosmological constant is positive, because the expansion of the Universe has been found to be accelerating in the 1990s, by observing distant supernovae (a feat for which Saul Perlmutter (1959-) Brian P. Schmidt (1967-) and Adam G. Riess (1969-) shared the Nobel prize for physics in 2011). Currently, the value of the cosmological constant is estimated to be:

L c² = 2.036 10^-35 / s²
L = 2.265 10^-52 / m²

That's the curvature of a sphere with a radius of about 7 billion light-years.

A mysterious dimensionless quantity can be obtained by multiplying the latter into the square of the Planck length:

5.917 10^-122 = 1 / 1.69 10¹²¹

The positiveness of the cosmological constant is (unfortunately) popularly described in term of what's dubbed dark energy (a positive vacuum energy with negative pressure, uniformly permeating empty space). This vocabulary doesn't add anything to the relativistic description of the large-scale structure of the Universe (in which the cosmological constant cannot vary, as its name implies). A discussion of the nature of dark energy would only be relevant beyond General Relativity, possibly within some future quantum theory of gravity...

Wikipedia : History of General Relativity | Einstein field equations | Einstein-Hilbert lagrangian (Hilbert, 1915)

Einstein's Famous Blunder (18:47) by Ed Copeland (Sixty Symbols, 2015年11月19日).
Gauged Six-dimensional Chiral Supergravity (6:10) by Ed Copeland (Sixty Symbols, 2011年04月20日).
Repulsive cosmological constant & "Big Rip" fad (1:47:36) by Leonard Susskind (2009年01月22日).

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(2009年07月07日) Motion of a Free-Falling Particle
Proper time is maximal along the spacetime path of freefall.

Matter tells space how to curve,
and space tells matter how to move.
John Archibald Wheeler (1911-2008)

Along a geodesic, the second-order variation of position vanishes:

d ² x^s + G s
mn dx^m dxⁿ = 0

One basic tenet of General Relativity is that gravity is part of the geometry (curvature) of spacetime. The spacetime path of a particle in free fall is simply a geodesic of spacetime; a path along which the elapsed proper time is extremal.

One is reminded of the principles of least time (Fermat, 1655) or least action (Maupertuis, 1744) which helped define the variational principles of mechanics (Lagrangian, Hamiltonian, etc.) at work here.

As "time" is just one of the spacetime coordinates, another arbitrary parameter l is used to describe a spacetime path Q(l) of fixed extremities along which the Lagrangian integrand is simply proportional to the interval of proper time :

[ -g _mn(Q) dx^m dxⁿ ] ^½ = [ -g _mn(Q) dx ^m dx ⁿ ] ^½ dl

vinculum vinculum

dl dl

This is a straight variational problem with a Lagrangian L(Q,V) proportional to

[ -g _mn(Q) v ^m v ⁿ ] ^½

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(2019年11月16日) Gravitational Lensing
Gravitation Bending Light

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The Strange Universe of Gravitational Lensing (13:31) by Matt O'dowd (2016年06月25日).

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(2009年08月01日) Relativistic Precession of Orbits
On the anomalous precession of the perihelion of Mercury (1915)

Newtonian gravity can be summarized as a relation between the mass density r and the Laplacian of the gravitational potential F (which is a negative quantity):

D F = 4 p G r

That static field can be described (for weak gravity and low speeds) by:

ds² = ( 1 - 2 F / c² ) [ dx² + dy² + dz² ] - ( 1 + 2F / c² ) c² dt²

Einstein himself used this approximation in 1915 (before he knew about the exact Schwarzschild metric) to explain the anomalous motion of the perihelion of Mercury (thus providing experimental support in favor of General Relativity ).

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Relativistic Precession Shift per Orbit (Einstein, 1920)

s = 24 p ³ a ² = 3 p R

vinculum vinculum

T ² c ² ( 1 - e ² ) a ( 1 - e ² )

For the orbit of Mercury :

• Semimajor axis : a = 5.790905 10¹⁰ m (= 0.3871 au).
• Eccentricity : e = 0.205630
• Period : T = 7.60053 10⁶ s (= 87.9691 days).
• Sun's pseudo-Schwarzschild radius: R = 2 G.S / c² = 2953.25 m.
  

R = 2953.25007703(19) m is known with ludicrous precision. (It would be the Sun's Schwarzschild radius if it was perfect;y spherical.)

The above yields the precession shift s in radians per revolution (of duration T). It's customary to give it in arcseconds per century instead. To obtain the result this way, multiply the above by factors equal to 1, knowing that a century is 3155760000 s and that p rad is 180° or 648000'' :

s = 3 R p
vinculum
a ( 1 - e ² ) rad
vinculum
T = 3 R 648000''
vinculum
a ( 1 - e ² ) T 3155760000 s
vinculum
century

This boils down to s = 42.9807'' / century. Using less precise raw date, Einstein famously rounded that to 43'' per century.

Sources of Apsidal Precession for Mercury's Orbit
(arcseconds per century)

Venus 277.42

Earth 90.89

Mars 2.48

Jupiter 153.99

Saturn 7.32

Uranus 0.14

Neptune 0.04

Oblateness of the Sun 0.03

Total (Newtonian) 532.33

Non-Newtonian 42.98

Total 575.31

Orbits in Strongly Curved Spacetime by John Walker

Motion of the Perihelion of Mercury by Albert Einstein (1920).

The Precession of the Perihelion of Mercury by David Eckstein (2013).

A simple model of the chaotic eccentricity of Mercury by G. Boué, J. Laskar & F. Farago2 (2012).

Vulcan, the planet that didn't exist (45:14) by Zepherus (2021年10月25日).

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Karl Schwarzschild 1873-1916
Karl Schwarzschild

(2009年08月01日) The Schwarzschild Metric (1915)
An exact solution due to Karl Schwarzschild (1873-1916)

In 1923, George David Birkhoff (1884-1944) proved that what Schwarzschild had described in 1915 is actually the only spherically symmetric static solution to Einstein's field equations. That unicity had been discovered in 1920 by Jørg Tofte Jebsen (1888-1922) but it wasn't promoted because of Jebsen's battle with turberculosis (in spite of C.W. Oseen's efforts).

ds² = (1-a/r)^-1 dr² + r² dW² - (1-a/r) c² dt²

where dW² = dq² + sin² q dj²

This represents the relativistic gravitational field around a (structureless) point of mass M if we let the so-called Schwarzschild radius be:

a = 2 G M / c²
Swift-footed Achilles and the Tortoise

Tortoise Coordinates :

As the radial parameter r is not directly proportional to the radial distance described by the above metric, it makes sense to use a parameter u which is. More precisely, we introduce a u as the radial distance to the event horizon when outside of it:

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Eddington-Finkelstein coordinates
Kruskal-Szekeres coordinates
Achilles & the Tortoise (Zeno).

Painlevé-Gullstrand coordinates (PG) :

This proposal is now of historical interest only. It was originally mistaken as an alternative spherically-symmetric solution to Einstein field equations, distinct from the Schwarzschild metric (contradicting Birkhoff's theorem).

Two noted early detractors of Einstein's theory made the same proposal independently: Paul Painlevé (in 1921) and Allvar Gullstrand (in 1922).

Painlevé (1863-1933) was a French mathematician who had turned to politics in 1906 (he served as prime minister of France in 1917 and 1925). Gullstrand (1862-1930) was a surgeon and ophthalmologist who became a member of the Royal Swedish Academy of Sciences in 1905. He disbelieved the theory of relativity and was instrumental in blocking the award of the Nobel prize to Einstein for it (Einstein got it in 1921 for the photoelectric effect instead) Gullstrand had received the 1911 Nobel prize in medicine for his studies of the optics of the human eye. The Gullstrand formula gives the optical power resulting from two optical systems separated from each other (e.g., eye and corrective lens).

Both argued that the existence of two possible fields for the same distribution of mass demonstrated the ambiguity or incompleteness of General Relativity. Einstein questioned the physical relevance of their proposed metric, involving a puzzling cross-term between spatial and time coordinates. The issue was settled, in 1933, by Georges Lemaître who showed how the controversial proposal was physically equivalent to the Schwarzschild solution, merely presented in a strange coordinate system.

Wikipedia : Schwarzschild metric (1915) | Lemaître coordinates (1932)
Eddington-Finkelstein coordinates | Kruskal-Szekeres coordinates | Painlevé-Gullstrand coordinates
Martin Kruskal (1925-2006) | George Szekeres (1911-2005)

The Phantom Singularity (18:04) by Matt O'Dowd (PBS Space-time, 2017年01月19日).
Singularities Explained (10:22) by Kelsey Houston-Edwards (PBS Infinite Series, 2017年01月19日).

What's on the Other Side of a Black Hole? (15:14) by Matt O'Dowd (PBS Space-time, 2020年03月31日).

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(2013年12月14日) The Vaidya metric
Gravity field outside a spherical body immersed in massless radiation.

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Wikipedia : Vaidya metric | Null-dust solution
Prahalad Chunnilal Vaidya (1910-2010)

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(2019年11月09日) van Stockum dust (Lanczos 1924, van Stockum 1937)
An artificial exact GR solution with closed timelike curves.

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Wikipedia : van Stockum dust | Dust solution | Fluid solution | Gödel metric (1949)
Cornelius Lanczos (1893-1974) | Willem Jacob van Stockum (1910-1944)

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meglovessims (Yahoo! 2007年08月11日) What is mass ?
Is mass a property of matter?

Mass can be defined in two different ways:

• Inertial mass. The more mass an object has, the more difficult it is to change its motion. You multiply mass by velocity to obtain momentum.
• Gravitational mass. The more mass an object has, the greater the force (called "weight") a given gravitational field exerts on it. Technically, you multiply mass by gravity to obtain weight.

The fact that both approaches define exactly the same thing is the so-called equivalence principle. It's a basic tenet of Einstein's General Relativity.

A distinction must be made between ordinary mass (which you may call "rest mass" if you must) and the above "relativistic mass", which is strictly proportional to the total energy E. Nowadays, people rarely use the concept of relativistic mass anymore, since the proportionality with E makes it look like a waste of an otherwise badly needed symbol (m).

Neither concept is reserved to particles of matter (fermions). Both properties can also be assigned (at least in some cases) to the force messengers (bosons). This is especially true for relativistic mass, which is associated to anything with nonzero energy. For example, a photon of frequency n has an energy hn and, therefore, a relativistic mass hn/c² (where h is Planck's constant). Photons have inertia and are deflected by gravity (and conversely cause some gravity). Yet, they have no proper mass; they cannot exist at rest. Any object of zero mass can only have nonzero energy if it travels exactly at the speed of light (c).

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(2005年08月21日) Unruh radiation, Unruh temperature, Unruh effect (1976)
An accelerated observer experiences a heat bath of photons.

Bill Unruh (b. 1945) of the University of British Columbia showed in 1976 that an observer submitted to an acceleration (or a gravitational field) g experiences a bath of photons whose temperature is proportional to g.

Unruh temperature T for an acceleration g

kT = g h / (4p²c) [ Any coherent units ]

T = g / 2p [ In natural units ]

The corresponding thermal radiation is due to the fact that, for an accelerated observer, there is an event horizon which may trap one of two paired particles in a particle-antiparticle creation. Unruh radiation is thus similar to the better-known Hawking radiation for black holes, which is described by the same formula (for Hawking radiation, g is the gravity on the black hole's event horizon).

Fulling-Davie-Unruh effect (1973, 1975, 1976)
Bill Unruh (1945-) | Paul Davies (1946-) | Stephen A. Fulling (1945-)

Videos : The black hole death problem (5:11) by Dianna Cowern (Physics Girl, 2016年02月02日).
The Ubruh effect (11:13) by Matt O'Dowd (PBS Space Time, 2018年04月04日).

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(2005年07月16日) Tensorial Form of Electromagnetism
The equations of electromagnetism have simple relativistic expressions.

Covariant Potential and the Faraday Tensor

The electromagnetic fields form a covariant antisymmetric tensor F which is the 4-dimensional rotational of the covariant potential A:

Covariant Electromagnetic Potential

A_n = ( -f/c, A_x, A_y, A_z ) = ( -f/c, A )

Covariant Faraday Tensor F = - Rot A

F_mn = A_n,m - A_m,n = A_n;m - A_m;n

bracket

F₀₀ F₀₁ F₀₂ F₀₃

F₁₀ F₁₁ F₁₂ F₁₃

F₂₀ F₂₁ F₂₂ F₂₃

F₃₀ F₃₁ F₃₂ F₃₃

bracket = bracket

0 -E_x /c -E_y /c -E_z /c

E_x /c 0 B_z -B_y

E_y /c -B_z 0 B_x

E_z /c B_y -B_x 0

bracket

bracket bracket bracket bracket

bracket bracket bracket bracket

In flat space (no gravity) the doubly-contravariant coordinates of F are:

bracket

F⁰⁰ F⁰¹ F⁰² F⁰³

F¹⁰ F¹¹ F¹² F¹³

F²⁰ F²¹ F²² F²³

F³⁰ F³¹ F³² F³³

bracket = bracket

0 E_x /c E_y /c E_z /c

-E_x /c 0 B_z -B_y

-E_y /c -B_z 0 B_x

-E_z /c B_y -B_x 0

bracket

bracket bracket bracket bracket

bracket bracket bracket bracket

Therefore, F_mn F^mn = 2 ( - E²/c² + B² ) which is proportional to the Lagrangian density compatible with the Hamiltonian energy density derived from the Poynting theorem, namely:

Electromagnetic Lagrangian Density

¹/₂ e_o ( E ² - c² B ² ) = - F_mn F^mn / 4m_o

How Special Relativity Makes Magnets Work (4:18) by Derek Muller and Henry Reich (2013年09月23日).

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(2009年08月05日) Kaluza-Klein Theory
Using a fifth spacetime dimension to explain electromagnetism.

The theory formulated by Theodor Kaluza (1885-1954) in 1919 and refined by Oskar Klein (1894-1977) in 1926 contains a remarkable idea which is still with us as an essential ingredient of modern string theory: Fundamental forces besides gravity may have a unified explanation in a framework where spacetime has more than 4 dimensions... This approach currently seems to be the most promising way to construct quantum theories compatible with gravity (in fact, quantum theories where gravity looks unavoidable).

Although the original 5-dimensional Kaluza-Klein theory did not reach its goal of providing a perfect explanation for electromagnetism, the core of that classical theory repays study. Here it is:

Consider a 5-dimensional spacetime obtained by adding a fifth dimension to the usual 4D spacetime considered so far (the fifth index is equal to 4). We keep the usual symbols for 4D quantities and primed symbols for their 5D counterparts. Greek indices run from 0 to 3, latin indices run from 0 to 4.

We assume the following relations (with ¶₄ g'_mn = 0 ) :

g'_mn = g _mn + A _m A _n g'_m4 = g'_4m = A _m g'₄₄ = 1

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History of String theory | Kaluza-Klein theory by Christopher N. Pope | Wikipedia : Kaluza-Klein

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(2007年08月09日) Harvard Tower Experiment (Pound & Rebka, 1959)
Demonstrating the gravitational redshift (using the Mössbauer effect ).

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Pound-Rebka experiment (1959) | Robert Pound (1919-2010) | Glen A. Rebka, Jr. (1931-)

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(2009年04月10日) Shapiro Delay (Irwin I. Shapiro, 1964)
Gravitational time dilation causes apparent delays in radar signals.

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PSR J1909-3744 (roundest known orbit in the universe, 2006)
Wikipedia : Shapiro Delay

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(2012年12月06日) Warp Drive (Alcubierre 1994, Zefram Cochrane 2063)
Contract space in front of yourself and expand it behind yourself.

Since this writer is supposed to pass away on 21 March 2035, any reference to the scientific accomplishments of Zefram Cochrane (2030-2117) is necessarily facetious.

On the hand, Miguel Alcubierre did work out a particular propagation of a spatial disturbance which allows faster than light (FTL) travel without ever violating the absolute local speed limit of 299792458 m/s. The distribution of matter-energy corresponding to Alcubierre's solution is then simply obtained from Einstein's field equation. It involves a total negative energy of the same order of magnitude as the total mass in the observable universe...

Negative energies are not ruled out (the Casimir effect implies the existence of negative energy between parallel plates in the real world). Therefore, some people have studied different configurations which duplicate the properties of the Alcubierre drive with a much lower need for negative energy. This includes Sonny White and his group at NASA...

Discovery Channel : Can We Travel Faster Than Light?

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(2016年08月25日) Relativistic Frame Dragging (Lense and Thirring, 1918)
On the orbital precession predicted by Josef Lense and Hans Thirring.

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Lense-Thirring effect | Josef Lense (1890-1985) | Hans Thirring (1888-1976) | Frame-dragging

Howard P. Robertson
Bob Robertson

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(2017年02月19日) Photon braking: Poynting-Robertson drag
Photons emitted from a star slow down orbiting dust.

In 1937, Bob Robertson ...

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Under favorable conditions, sunspots can be observed at sunrise with the naked eye. That was reported by the Chinese before Charlemagne's reign.

Starting in 1610, sunspots were independently observed by several early telescope users, including Galileo, his nemesis Christoph Scheiner and Thomas Harriott (December 1610).

Johann Fabricius of Wittenberg famously made his own first observation of sunspots on March 9, 1611 and showed them immediately to his father, David Fabricius. Johann Fabricius went on to track sunspots from one day to the next and could infer that the Sun was rotating at a rate of about one revolution per month. He published the first report on the new findings in the Summer of 1611 ( De Maculis in Sole Observatis ). This remained unnoticed for a while.

Why is the Sun Slowing Down? (5:45) by Derek Muller (Sciencium, 2017年02月16日).

Poynting-Robertson-like Drag at the Sun's Surface (2017年02月03日)
Ian Cunnyngham, Marcelo Emilio, Jeff Kuhn, Isabelle Scholl, and Rock Bush Phys. Rev. Lett. 118, 051102.

Sunspots (The Galileo Project) | Poynting-Robertson effect (Poynting 1903, Robertson 1937)
John Henry Poynting (1852-1914) | Howard Percy "Bob" Robertson (1903-1961)