The general theory of relativity can be conceived onlyas a field theory. It could not have[been] developed if onehad held on to the view that the real world consists of material pointswhich move underthe influence of forces acting between them. AlbertEinstein, in his last scientific paper (December 1954).
Albert Einstein remarked thatthe force exerted by gravity on an object [which we call the weight of that object] is strictly proportional to its inertialmass, just likethe 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 consequencesof putting gravitational forces and inertial forces on the same footing.
The only difference between gravity and an ordinary fictitious force fieldis that the former cannot (usually) be reduced to a mere artifact ofcoordinate motion. So, with gravity, we no longer have the luxuryof going back at will to an "inertial frame" where physical laws are simpler. Instead, we're stuck with a system of coordinates correspondingto 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 descriptionof how gravity isproduced 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.
(2021-02-16) 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 surfaceof a spherical star (of mass M and radius R) gets deflectedby the following small angle:
= 2 M G / R c2
Einstein himself obtained this non-relativistic relation in 1911 using his principle of equivalence before he worked out thefull 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 otainedin two different ways:
At perihelion (when the light ray grazes the surface of the Sun, at a distance R from thecenter) 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:
= 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.
= 4 M G / R c2
Louis Vlemincq (2005-07-25; e-mail) Does the Harress-Sagnac effect contradict General Relativity ?
The Sagnac effect is simply the observation thattwo beams of light circling the same rotating loop in oppositedirections 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 fromthe other before light returns to it).
In the main, the version of theSagnac effect whichinvolves mirrors rather than fiber optics is nonrelativistic. In ourintroduction to the Sagnac effect,we've shown that special relativity implies that a Sagnac apparatusmade from fiber optics worksexactly like a mirrored oneenclosing the samesurface (regardless of the refractive index n of the optical cable used).
For some obscure reason, the Sagnac effect has been toutedas "alternative science". It's not. In fact, the Sagnac effect has been providing a reliable solid-statesubstitute for gyroscopesaboard aircrafts for over 30 years.
ASagnac 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 involvedin general relativity (GR).
The example of the rotating disc is what convinced Einstein himselfthat Euclidean geometry was inadequate in a general coordinate system wherean observer at rest would see masses accelerate from either of twoequivalent causes: gravitational fields or nonuniform motion (with respectto a local Lorentzian "inertial" system).
We've establishedelsewhere the following expressionfor the time lag in therespective returns of two light beams traveling in oppositite directionsaround a circular loop of radius R,rotating around its axis at a rate .
t =
4 R 2
c 2 2 R 2
This expression is valid for an inertial [nonrotating] observer who does not movewith respect to the loop's center of rotation. The main reason for the observed nonzero lag time t is that each beam must travel a different distance to reach the which moves with the loop. A careful analysis with fiber opticsreveals that t does not depend on the index of refraction (n) and is the same for amirrored apparatus as well (n = 1).
It's enlightening to ponder the above expression,which we may rewrite:
t =
( 4 / c2)
1 (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.
is the loop's vectorial surface,anaxial vector which depends not only on theconventional orientation of space but also on which direction is chosen as positive to travel around the loop. In Euclidean geometry, maybe defined by a contour integral around the oriented loop (C+).
= C+r dr
explicitly fora circle of radius R, with the following parametric equations (0 < < 2).
x = a + R cos ; y = b + R sin ; z = c
Now, the denominator in the above looks like a relativistic correction (indeed it is) which we may discard at first
Sagnac Time Lag (observer tied to the loop)
t' =
4
c 2
(2005-07-29) Arigid motion is a state ofequilibrium, which can change only so fast.
In classical mechanics, a solid is a body whose parts always remain atthe same distances from each other, in what's called rigid motion. In such a motion there must be a rotation vector which ties the velocities of any pair A andB of the solid's points, via the followingrelation ( isanaxial vector whose sign depends on space "orientation").
vAA = vBB
This is only a good approximation to physical reality if any change in the velocityof a point is somehow made known instantly throughout thesolid 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 thespeed of sound within the solid. Loosely speaking, a change in rotation which starts at the axis of rotationwill 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 rotationquite violently, but its ultimate state is an unchanging state of equilibrium which may beworth studying. (Even so, it's fallacious to consider a solid with parts moving faster than light !)
(2009-07-25) A gentle introduction ( = 1,2) to tensor notations ( = 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 anynonzero 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).
A symbolwith an index appearing as a superscript isdifferent from the same symbol withthe 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 beits [contravariant] coordinates in that basis:
dV = dv1ê1 + dv2ê2
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 alinear combination) for the dual basis consisting oftwo 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 = ij ( equal to 1 if i = j and zero otherwise)
The following relations hold:
dV
=
dv1ê1 + dv2ê2
=
(dV.ê1) ê1 + (dV.ê2) ê2
=
dv1ê1 + dv2ê2
=
(dV.ê1) ê1 + (dV.ê2) ê2
|| dV || 2
=
dv1 dv1 + dv2 dv2
All this is best expressed by introducing themetric tensor g (whichdefines the dot product and, hence, the notion of length). g is symmetric ( g12 = g21 ) because the dot product is commutative ( ê1 . ê2 = ê2 . ê1 ).
ê1
=
g11ê1 + g12ê2
=
(ê1. ê1)ê1 + (ê1. ê2)ê2
ê2
=
g21ê1 + g22ê2
=
(ê2. ê1)ê1 + (ê2. ê2)ê2
By definition, the square of the length of dV is the dot product dV . dV :
|| dV || 2
=
( dv1ê1 + dv2ê2) . ( dv1ê1 + dv2ê2)
=
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 that appears in this last expression is the multiplicative inverse of g whichexpresses the reciprocal linear relations, namely:
ê1
=
g11ê1 + g12ê2
=
(ê1. ê1)ê1 + (ê1. ê2)ê2
ê2
=
g21ê1 + g22ê2
=
(ê2. ê1)ê1 + (ê2. ê2)ê2
The following relation holds in any number of dimensions:
To draw a nice picture where the circle of unit radiusdoes look like a circle, we use an orthonormal grid in which:
ê1 = ( 1, 0 ) ê2 = ( -1, 2 )
The metric tensor ( g ij = êi . êj ) is :
g ij
=
1
-1
-1
5
Its inverse is:
g ij
=
5
1
1
1
Therefore:
ê1
=
(5ê1 +ê2)/4
=
( 1, ½ )
ê2
=
(ê1 +ê2)/4
=
( 0, ½ )
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, whichgives the shape of the unit circle (orange) as a specificcartesian equation in that basis. The reciprocal basis vectors (red) are also specific linearcombinations of the original vectors which depend only onthe metric tensor...
Summary :
In a metric space, there's only one kind of vector, which may bespecified either by its contravariant coordinates or its covariant coordinates :
V = V1ê1 + V2ê2 = V1ê1 + V2ê2
The metric tensor and its inverse can be used toswitch back and forth between the contravariant and the covariant representations of any vector:
Vi = j g ij V j and V i = j g ij Vj
The backdrop of general relativity is essentially a generalization of this tofour dimensions with a metric of signature + + + (as opposed to + + for the Euclidean planediscussed above) whichsingles out the particular dimension of time. Technically speaking, relativistic spacetime is a Lorentzian manifold, a particular case ofsemi-Riemannianmanifold.
(2009-08-07) 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'
We use the "same basis" as before:
ê1 = ( 1, 0 ) ê2 = ( -1, 2 )
The metric tensor ( g ij = êi . êj ) is :
g ij
=
1
-1
-1
-3
Its inverse is:
g ij
=
3
-1
-1
-1
Therefore:
ê1
=
(3ê1ê2)/4
=
( 1, - ½ )
ê2
=
(ê1ê2)/4
=
( 0, - ½ )
(2009-08-05) What tensors really are.
By definition, the scalars of a vector space are its tensors of rank 0.
In anyvector space,a linear function which sends a vector to a scalar may be calleda covector. Normally, covectors and vectors are different types of things. (Think of the bras and kets ofquantum mechanics.) However, if we are considering only finitely many dimensions,then the space of vectors and the space of covectors have thesame number of dimensions and can therefore be put in a linear one-to-one correspondence with each other.
Such a bijectivecorrespondence is called a metric and is fully specified by a nondegenerate quadratic form, denoted by a dot-product ("nondegenerate" precisely means that the associated correspondenceis bijective).
Once a metric is defined, we are allowed to blur completelythe distinction betweenvectors and covectors as they are now in canonical one-to-onecorrespondence. 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 nth-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).
(2009-07-29) Bases in which a given metric tensor has its simplest expression.
In the previousintroductory article, we definedthe metric tensor with respect to a particular basisin terms of a known ordinary euclidean dot product:
g ij = ê i . ê j
From that metric tensor alone, wecomputed reciprocal vectors satisfying:
ê i . ê j = i j
It turns out that this can always be doneif 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 onthe 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.)
One easy way to determinethe signature of a given metric tensor (or any hermitianmatrix, actually) is to use Descartes'rule of signs (1637) on its characteristic polynomial (whose roots are all real).
ê i. ê j
=
g ij = ijalways (by definition of thereciprocal vectors).
ê i. êj
=
gij = ij in an orthonormal basis only.
For a nondegenerate metric,ij = 0 when i j whereas ii = 1.
(2009-07-21) 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 coordinates :
The value of each coordinate y is a function of the event itselfand is, therefore, a function of all four x-coordinates. Thedifferential of each y-coordinate is thusa linear combination of the differentials of the four x-coordinates. By definition, the coefficients of those linear combinations are knownas partial derivatives :
d y =
y
dx0 +
y
dx1 +
y
dx2 +
y
dx3
x0
x1
x2
x3
In relativistic tensor calculus, such sums are rarely written outexplicitly. Instead, the Einstein summation convention is used, which states that a multiplicative expressionwhere an index occurs twice (once downstairs and once upstairs) denotes the sum of 4 terms where that index takes on allvalues from 0 to 3. Thus, the above sum is equivalent to:
d y =
y
d x
x
If the four coordinates of a vectorialquantity V obey the transformation rulesthat we just established for an infinitesimal spacetime displacement,they are called contravariant coordinates and bear superscripted indices:
V[y] =
y
V[x]
x
Instead of using different sets of names, we mayunderscore 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 = x V
Each vector ê of a local reference frameis identified with a lower index from 0 to 3, to conform tothe standard restriction,which says that a summation index must appear once as a subscriptand once as a superscript:
Expansion of a vector using contravariant coordinates
V = Vê = V0ê0 +V1ê1 +V2ê2 +V3ê3
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 denotean object with components that are not made explicitby 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 byother schemes besides the aforementioned contravariant linear combinations. Indeed, one such scheme is the covariant viewpoint which we are aboutto describe (both aspects become interchangeable in ametric space).
Gradients and other coordinates :
Transformation of Covariant Coordinates
V = x V
(2009-07-21) and its inverse g Lowering or raising indices.
The spacetime interval (squared) is gdxdx
(2009-07-25) A dualis obtained by switching all indices (andcomplex conjugation).
(V)* = (V*) (V)* = (V*)
(2009-07-23) Derivativeswith respect to contravariant or to covariant coordinates.
Loosely speaking, a lower index at the denominator becomesan upper index for the overall ratio, and vice-versa.
Thus, the derivative with respect to a contravariant coordinatecarries a lower index whereas the derivative with respect to a covariant coordinatecarries an upper index. Those two operatorsapplied to are respectively denoted:
= =
x
= =
x
(2009-07-30) 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.
The so-called coefficients of affine connection are simplythe coordinates of the partial derivatives of the basis vectors. They are better known as Christoffel symbols (orgammas) and may be defined as follows:
= ê.ê
Since ê .ê = is a constant,its derivatives vanish and we have:
= ê.ê
It's best to maintain the order of the downstairs indices (the differentiation index () should be placed last ) although that order is irrelevant in Einstein's [standard] General Relativity because of the symmetry induced by theequivalenceprinciple. A few authoritative references support that convention:
Misner et al. (1973) Equation 8.19a, page 209. [Strongly!]
Some reputable authors (includingWeinberg andWald) 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 theChristoffel 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
(2009-07-29) Introducing the covariant derivatives
Covariant derivatives are due toGregorioRicci-Curbastro (1853-1925) who invented most of tensor calculusbetween 1884 and 1894 (Delle derivazione covariante e contravariante, Padova, 1888). In1900,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 ofCarl Friedrich Gauss (1777-1855), Bernhard Riemann (1826-1866) and ElwinChristoffel (1829-1900).
times its diameter ! Equating inertial accelerations and gravitational fields (his principle of equivalence) Einstein suspected that gravity might be related toa local disturbance in the metric features of spacetime...
A tensor field is a function (linear or not) mapping a spacetime point m to some tensorT of rank n. The linear function mapping aninfinitesimal (vectorial) displacement dm to the corresponding variation of T is thus a tensor of rank n+1 which is denoted T . By definition:
T ( dm ) = d [T (m ) ]
Loosely speaking, that's also equal to T (m + dm ) T (m )
The covariant derivative is to the absolute differentiation of a tensor T what the partial derivative is to thedifferentiation of a scalar f.
T = êT
d f = dxf
Formally, can thus be defined by dotting into ê
= ê.
A vector is really a linear combination of basis vectors which maywell change as the differentiation variable varies. Therefore, theproduct rule fully applies (that's similar to the wayrigid motion brings about a rotation vector ) and we obtain:
(V ) ( Vê)
=
V
ê
+
Vê
x
So,
êVê)
=
V
êê
+
(êê) V
x
=
V
êê
+
(êê) V
x
Since and have the same effect on basis vectors,what appears in the last bracket is actually the nabla operator = ê applied to ê. The coordinates of that are theChristoffel symbols introduced above:
ê =
êê
ê =
ê
The latter equation implies the former, which we plug into the above to obtainthe following expression for the coordinates of the covariant derivative of a vector:
Covariant derivative of a vector
V V =
V
V
x
The covariant derivative of a tensor of rank n entails a sum of n+1 terms:
T =
T
T
T
x
U =
U
U
U
U
x
If upper indices are used, the coordinates of contravariant derivatives obeya similar rule with the same symbols, but different summationsand opposite signs:
V V =
V
V
x
Here's one example with mixed indices (one upstairs, two downstairs):
U =
U
U
U
U
x
This simple statement summarizes 256 formulas,with 13 terms each...
The tensorial operator obeys the standard rules forraising and lowering ofindices. This is consistent with its two equivalent (dual) expressions:
= ê = ê
Indeed, by dotting everything into ê we obtain:
ê. = = g
We may also obtain expressions for contravariant derivatives ab initio :
(V ) ( Vê)
=
V
ê
+
Vê
x
So,
êVê)
=
V
êê
+
(êê) V
x
=
V
êê
+
(ê) V
x
The last term is just what we have in the same circumstances for covariant derivatives but we must now express itover a different tensorial basis (matching that of the first term) to obtain proper component-wise relations:
ê =
êê
=
gêê
We may thus introduce the following not-so-common notation to simplify theexplicit expressions for contravariant derivatives given below:
=
g
Contravariant derivatives of a vector
V V =
V
V
x
V V =
V
V
x
This is just for completeness... Those explicit formulas for contravariant derivativesare rarely used, if ever. They can be generalized to tensors of higher ranks by using the same patterns ascovariant derivatives.
(2009-10-21) 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 toprove that the ordinary derivatives of a tensor forma 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 thethe first and second derivativesof one set of coordinates with respect to the other set. It is the presence of second derivatives which indicates thatChristoffel symbols are not tensors. However, the symmetry of those second derivatives make them vanish from thetransformation 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 calledthe torsion of spacetime.
Q = ½ (
)
Formally,
(2009-10-15) A postulate implying the symmetry of Christoffel symbols.
TheChristoffel symbolsdo not form a tensor, but the followingquantity is a proper antisymmetric tensor called Cartan torsion (or Cartan tensor ) :
Q = ½ (
)
The principle of equivalence postulated in Einstein's general theory of relativity implies that spacetime is torsion-freeas 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 otherframe of reference as well, which means that Christoffel symbols are always symmetrical withrespect to their two lower indices.
If such a torsion-free spacetime ismetric-compatible, thenthe Christoffel symbols are functions of the metric coefficientsand their first derivatives:
Torsion-free Christoffel symbols :
=
g ( g g g)
Although the Christoffel symbols do not form a proper tensor, we may still introducethe following notation which will clarify the discussion:
In the torsion-free case, we may use the symmetry of with respect to its last two indicesand the addition of those three equations yields:
g g g = 2
The advertised result is obtained by multiplying both sides into g
Note that,conversely, the above formula only holds in the torsion-free case (as it does give Christoffel symbols that are symmetrical with respectto their last two indices). It also implies themetriccompatibility which was used to derive it (the reader may want to checkalgebraically that thecovariant derivatives ofthe metric tensor vanish when the Christoffel symbols have those advertised values).
(2009-10-23) What if the torsion Qis a totally antisymmetric tensor...
With nonzero torsion in ametric-compatible geometry, the final summation in theabove proof yields thefollowing equation:
g g g = + + 2 [ Q +Q ]
It is tempting to consider the case where the square bracket vanishes. Since Q is already known to be antisymmetricwith respect to its last two indices, this additional antisymmetrywould make it a totally antisymmetric tensor.
In that case, the formula of the previous section just givesthe symmetric part of the Christoffel symbolsand, therefore, its generalization becomes:
=
g ( g g g) + Q
Conversely, such connection coefficients involving a totally antisymmetrictorsion Q describe ametric compatible affine geometry.
g ij; = Qig j Qjg i = Qji Qij = 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 vectorthe (essentially unique) totally antisymmetric tensor of rank 4. In other words, this kind of torsion can be described by avector field...
(2009-08-15) ij , ijk , ijkl , ijklm , etc. Antisymmetric with respect to any pair of indices.
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 ak-vector. Hodge duality is a linear bijection between k-vectors and (n-k)-vectors. W. V. D. Hodge (1903-1975).
(2009-07-28) The covariant derivatives of the metric tensor vanish.
Ricci's theorem means that covariant differentiation commutes with theraising or lowering of indices. This result is dubbed metric compatibility and can be construed as the fundamental theorem of tensor calculus. Ricci established it in 1884.
Metric Compatibility
g = 0
This is virtually an axiom nowadays (like thePythagorean theorem has become an axiom defining distancein modern Cartesian geometry). Metric compatibility demands that the dot product of twoparallel-transported vectors remain constant.
The situation is simpler than it sounds. One elementary way to visualize it is to consider the special case ofa two-dimensional curved surface in Euclidean three-dimensionalspace... If the quadratic form corresponding tothe metric tensor on that surface actually describes the 3D Euclidean metric, thenit follows that it's invariant in the absolute sense underlyingcovariant differentiation.
The same would hold true for a curved "surface" of any dimensionembedded in any "straight" space of higher dimension (endowed with a coordinate system where the higher-dimensionalmetric tensor is constant).
Once this remark is made, the expression of theChristoffel symbols in term of themetric coefficients can be obtained and we can forgetabout the crutch (or luxury) ofbeing able to reason in a higher-dimensional space witha simpler structure.
Although that simpler encompassing structure may not exist, therelation between Christoffel symbols and metric coefficientswhich is derived from that mere possibility isgiven the name of metric compatibility.
In a freely falling cartesian frame of reference,the components of the metric tensor are constant and the Christoffelsymbols vanish. Thus, the covariant derivatives of the metric tensor vanish in thisframe of reference and, therefore, in any other.
(2009-07-31) 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 thethecommutator ofcovariant derivatives as follows:
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):
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 = R
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.
(2009-08-08)
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 classmateGregorio Ricci-Curbastrowho 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 + R + R = 0
A contracted version holds for the Ricci tensor (: multiply by g).
Contracted Bianchi identity:
R R R = 0
By contracting this with respect to the indices and, we obtain:
R R R = 0
The Ricci scalar R = R appears in the first term. The second and third terms happen to be equal. So, the whole relation boils down to:
R 2 R = 0
That key relation establishes that the following tensor, introduced by Einstein,has a vanishing divergence (i.e., G = 0 ).
Definition of Einstein's Tensor:
G = R½ g R
Besides the metric tensorg itself, the Einstein tensorG turns out to be the only divergence-free second-rank tensor that can be builtfrom 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 twofundamental constants of nature, and G) between the three prominent divergencelesssecond-rank tensors g, G and T. The third of those is the stress tensorT, discussednext, whose lack of divergence expresses the conservationof energy and momentum.
(2020-09-17) Traceless component of the Riemann tensor.
(2009-07-31) 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 areall conserved quantities. The 4-dimensional flow of that quadrivectoris a tensor of rank 2 whose spatial components havethe dimension of a pressure; it's called the stress tensor. (Also called stress-energy or energy-momentum-stress.)
(2005-08-22) Presented to the Prussian Academy of Science on November 25, 1915.
The elements of the stress tensor T are in units ofenergy 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) thenthe metric tensor is dimensionless and the intrinsic curvatures arehomogeneous to the reciprocal of a surface area. So is the cosmological constant.
The Cosmological Constant
Einstein introduced it in 1917 for the wrong reasons, as he remarked that such a cosmological term would allows the Universeto 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) thatthe Universe is not static (it's expandingat arate of about 70 km/s/Mpc). Second, a Universe obeying Einstein's complete equationwould not durably remain in equilibrium (the slightest perturbation in the distribution of its contents wouldget amplified and result in a nonstatic universe). A zero cosmological constant was compatible with observations madeduring the lifetime of Einstein. That's why he once saidthat introducing it was "the worst mistake of [his] life".
With hindsight, the discussion should have focused on the actual valueof 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:
c2 = 2.036 10-35 / s2 = 2.265 10-52 / m2
That's the curvature of a sphere with a radius of about 7 billion light-years.
A mysteriousdimensionless 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 describedin 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 descriptionof 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...
(2009-07-07) Proper time is maximal along the spacetime path of freefall.
Matter tells spacehow 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 2x +
dxdx = 0
One basic tenet of General Relativity is that gravityis part of the geometry (curvature) of spacetime. The spacetime path of a particle in free fall is simplya geodesic of spacetime; a path along which the elapsedproper time is extremal.
As "time" is just one of the spacetime coordinates, another arbitrary parameter is used to describe a spacetime path Q() of fixed extremitiesalong which the Lagrangian integrand is simply proportional to the interval of proper time :
-g(Q) dxdx = -g(Q)
dx
dx
d
d
d
This is a straightvariational problem with aLagrangian L(Q,V) proportional to
[ -g(Q) vv ]
(2019-11-16) Gravitation Bending Light
(2009-08-01) On the anomalous precession of the perihelion of Mercury (1915)
Newtonian gravitycan be summarized as a relation between the mass density and the Laplacian of the gravitationalpotential (which is a negative quantity):
= 4 G
That static field can be described (for weak gravity and low speeds) by:
Einstein himself used this approximation in 1915 (before he knew about the exactSchwarzschild metric) to explain the anomalous motion of the perihelion of Mercury (thus providing experimental support in favor of General Relativity ).
Relativistic Precession Shift per Orbit (Einstein, 1920)
Sun's pseudo-Schwarzschild radius: R = 2G.S / c2 = 2953.25 m.
R = 2953.25007703() m is known with ludicrousprecision. (It would be the Sun's Schwarzschild radius if it was perfect;y spherical.)
The above yields the precession shift 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 rad is 180° or 648000'' :
=
3 R a ( 1 e 2)
rad T
=
3 R 648000'' a ( 1 e 2) T
3155760000 s century
This boils down to = 42.9807'' / century. Using less precise raw date, Einstein famously rounded that to 43'' per century.
Sources ofApsidalPrecession for Mercury's Orbit (arcseconds per century)
In 1923,George David Birkhoff (1884-1944) proved that what Schwarzschild had described in 1915 is actuallythe only spherically symmetric static solution toEinstein's field equations. That unicity had been discovered in 1920 byJørg Tofte Jebsen(1888-1922) but it wasn't promoted because of Jebsen's battle with turberculosis (in spite ofC.W. Oseen's efforts).
ds2 = (1-a/r)-1 dr2 + r2 d2 (1-a/r) c2 dt2
where d2 = d2 + sin2 d2
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
Tortoise Coordinates :
As the radial parameter r is not directly proportional to theradial distance described by the above metric, it makes sense to usea parameter u which is. More precisely, we introduce a u as the radial distance to the event horizon whenoutside of it:
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 solutionto Einstein field equations, distinct from the Schwarzschild metric (contradictingBirkhoff's theorem).
Two noted early detractors of Einstein's theory made the same proposal independently: Paul Painlevé (in 1921) andAllvar Gullstrand (in 1922).
Both argued that the existence of two possible fields for the same distribution of massdemonstrated the ambiguity or incompleteness of General Relativity. Einstein questioned the physical relevance of their proposed metric, involvinga 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 tothe Schwarzschild solution, merely presented in a strange coordinate system.
(2013-12-14) Gravity field outside a spherical body immersed in massless radiation.
(2019-11-09) An artificial exact GR solution with closed timelike curves.
meglovessims (Yahoo!2007-08-11) 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 otherwisebadly 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 has an energy h and, therefore, a relativistic mass h/c (where h isPlanck'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 thespeed of light (c).
(2005-08-21) 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 photonswhose temperature is proportional to g.
Unruh temperature T for an acceleration g
kT
=
g h / (42c)
[ Any coherent units ]
T
=
g / 2
[ In natural units ]
The corresponding thermal radiation is due to the fact that, for an accelerated observer,there is anevent horizonwhich 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).
(2005-07-16) 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 = ( /c, Ax, Ay, Az ) = ( /c,A )
Covariant Faraday TensorF = Rot A
F = AA = AA
F00
F01
F02
F03
F10
F11
F12
F13
F20
F21
F22
F23
F30
F31
F32
F33
=
0
-Ex/c
-Ey/c
-Ez/c
Ex/c
0
Bz
-By
Ey/c
-Bz
0
Bx
Ez/c
By
-Bx
0
In flat space (no gravity) the doubly-contravariant coordinatesof F are:
F00
F01
F02
F03
F10
F11
F12
F13
F20
F21
F22
F23
F30
F31
F32
F33
=
0
Ex/c
Ey/c
Ez/c
-Ex/c
0
Bz
-By
-Ey/c
-Bz
0
Bx
-Ez/c
By
-Bx
0
Therefore, FF = 2 (E2/c2B2) which is proportional to the Lagrangian density compatible withthe Hamiltonian energy density derived from thePoynting theorem, namely:
Electromagnetic Lagrangian Density
1/2o ( E 2 c2B 2 ) = FF/ 4o
(2009-08-05) Using a fifth spacetime dimension to explain electromagnetism.
The theory formulated byTheodor Kaluza (1885-1954)in 1919 and refined byOskar Klein (1894-1977) in 1926contains a remarkable idea which is still with us as an essential ingredientof modernstring theory: Fundamental forcesbesides gravity may have a unified explanation in a framework where spacetimehas more than 4 dimensions... This approach currentlyseems to be the most promising way to construct quantum theories compatiblewith gravity (in fact, quantum theories where gravity looks unavoidable).
Although the original 5-dimensional Kaluza-Klein theory did notreach its goal of providing a perfect explanation forelectromagnetism, 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 their5D counterparts. Greek indices run from 0 to 3, latin indices run from 0 to 4.
We assume the following relations (with g'mn = 0 ) :
g' = g + AA g' = g' = A g' = 1
(2007-08-09) (Pound & Rebka, 1959) Demonstrating the gravitational redshift (using the Mössbauer effect ).
(2009-04-10) (Irwin I. Shapiro, 1964) Gravitational time dilation causes apparent delays in radar signals.
(2012-12-06) (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 propagationof a spatial disturbance which allows faster than light (FTL) travelwithout ever violating the absolute local speed limit of 299792458 m/s. The distribution of matter-energy corresponding to Alcubierre's solutionis then simply obtained fromEinstein's field equation. It involves a total negative energy of the same order of magnitudeas the total mass in the observable universe...
Negative energies are not ruled out (the Casimir effect implies theexistence of negative energy between parallel plates in the real world). Therefore, some people have studied different configurations whichduplicate the properties of the Alcubierredrive with a much lower need for negative energy. This includes Sonny White and his group at NASA...
(2016-08-25) (Lense and Thirring, 1918) On the orbital precession predicted by Josef Lense and Hans Thirring.
(2017-02-19) Photons emitted from a star slow down orbiting dust.
In 1937, Bob Robertson ...
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 nemesisChristoph Scheiner and Thomas Harriott (December 1610).
Johann Fabricius of Wittenberg famously made his own first observation of sunspotson 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 thatthe Sun was rotating at a rate ofabout 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.
