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Final Answers
© 2000-2014   Gérard P. Michon, Ph.D.

General Relativity

The general theory of relativity can be conceived only as a field theory.
It could not have
[been] developed if one had held on to the view   Gregorio Ricci-Curbastro 
1853-1925  Albert Einstein 
1879-1955
that the real world consists of material points which move 
 under the influence of forces acting between them.

Albert Einstein,  in his last scientific paper (December 1954).

Related articles on this site:

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.
General Relativity from Special Relativity Using Tetrads  by  Will M. Farr.
Riemannian Geometry  (PHY3033, University of Hong-Kong)
Einstein-Cartan theory  by  Heiko Herrmann  (2004).
Einstein-Cartan theory  by  Andrzej Trautman  (arXiv, June 2006).
Cartan-Einstein Teleparallelism  (TP)  by  Jose 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

Books :

Videos :

Stephen Hawking's Universe
Episode 1:  Seeing is Believing  [  1  |  2  |  3  |  4  |  5  ]
Episode 2:  The Big Bang  [  1  |  2  |  3  |  4  |  5  ]
Episode 3:  Cosmic Alchemy  [  1  |  2  |  3  |  4  |  5  ]
Episode 4:  On the Dark Side  [  1  |  2  |  3  |  4  |  5  ]
Episode 5:  Black Holes & Beyond  [  1  |  2  |  3  |  4  |  5  ]
Episode 6:  Answer to Everything  [  1  |  2  |  3  |  4  |  5  ]
Sidney Colemean, Fay Dowker, Alan Guth, Andrei Linde,
Lee Smolin, Michio Kaku, Ed Witten, Neil Turok...

General Relativity Primer, three-hour seminar by  Sean M. Carroll  (Caltech)
XXXIII SLAC Summer Institute 2005transparencies | video 1 | video 2 | video 3

Einstein's Theory of General Relativity  by Leonard Susskind  (Fall 2008, Stanford University continuing education)  1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | iTunesU
General Relativity  (Susskind, Fall 2012)  1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

 
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General Theory of Relativity

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 (1879-1955) in 1921, 
 photographed by Ferdinand Schmutzer (1870-1928)
Albert Einstein  (1921)
 
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.


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

Mann muss immer generalizierenEdward "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  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).

 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 - (r)2 / c2

Note the bold type indicating  vectorial  quantities, defined as follows:

  • 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   òC+   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  explicitely 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

 Come back later, we're
 still working on this one...

Sagnac Time Lag  (observer tied to the loop)
Dt'     =       W . S
vinculum
c 2


(2005-07-29)   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").

vA -  W ´ A     =     vB -  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  tranverse  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 !)


(2009-07-25)   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   =   dij     ( equal to  1  if  i = j  and zero otherwise)

In three dimensions, crystallographers should recognize those as Ewald's "reciprocal basis vectors", which define the reciprocal lattice  (French: réseau réciproque )  associated to a Bravais lattice (named after Auguste Bravais (1811-1863, X1829) who held the chair of physics at Polytechnique from 1845 to 1856).

The following relations thus hold:

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  gmn  (which defines the dot product and, hence, the notion of length).  gmn  is symmetric  ( g12 = g21 )  because the dot product is commutative  ( ê. ê2  =  ê. ê).

  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  gmn  that appears in this last expression is the  multiplicative inverse  of  gmn  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    =   dij       (Kronecker's delta symbol)

 Dual Bases  

Example :

To draw a nice picture where the circle of unit radius does look like a circle, we use an  orthonormal  grid in which:

ê1 = ( 1, 0 )       ê2 = ( -1, 2 )

The metric tensor  ( g ij = ê. ê)  is :

[  g ij  ]   =   bracket
bracket
bracket
 1  -1  bracket
bracket
bracket
-1   5 
Its inverse is:  
[  g ij  ]   =  ¼   bracket
bracket
bracket
 5   1  bracket
bracket
bracket
 1   1 

Therefore:

ê1   =   (5 ê1 + ê2 )/4   =   ( 1, ½ )
ê2 = ( ê1 + ê2 )/4 = ( 0, ½ )

 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:

Vi   =   å j   g ij  V j         and         V i   =   å j   g ij  Vj

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.


(2009-08-07)   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'

 Dual Bases  

We use the "same basis" as before:

ê1 = ( 1, 0 )       ê2 = ( -1, 2 )

The metric tensor  ( g ij = ê. ê)  is :

[  g ij  ]   =   bracket
bracket
bracket
 1  -1  bracket
bracket
bracket
-1  -3 
Its inverse is:  
[  g ij  ]   =  ¼   bracket
bracket
bracket
 3  -1  bracket
bracket
bracket
-1  -1 

Therefore:

ê1   =   (3 ê1 - ê2 )/4   =   ( 1, - ½ )
ê2 = ( - ê1 - ê2 )/4 = ( 0, - ½ )

 Come back later, we're
 still working on this one...


(2009-08-05)   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  vectors  or tensors of rank 1  (there'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)   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   =   di 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).

 Come back later, we're
 still working on this one...

ê i . ê j  =  g ij   =   d ij     always  (by definition of the reciprocal vectors).
ê i . ê j = g ij   =   h ij     in an  orthonormal   basis  only.

For a nondegenerate metric, hij = 0  when  i ¹ j  whereas  hii = ± 1.

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


(2009-07-21)   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   =     ym   dx0   +   ym   dx1   +   ym   dx2   +   ym   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 explicitely.  Instead, the  Einstein summation convention  is used, which states that a multiplicative expression where an index occurs  twice  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   =     ym   d x n
vinculum
xn
The Einstein summation convention applies recursively:  Therefore, 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]   =     ym   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
Vm   =   n xm  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

Note that 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
Vm   =   n xm  Vn

 Come back later, we're
 still working on this one...


(2009-07-21)   The  metric tensor  gmn  and its inverse  gmn 
Lowering or raising indices.

The spacetime interval (squared) is   gmn dxm dxn

 Come back later, we're
 still working on this one...


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

(Vm )*   =   (V*) m                 (V m )*   =   (V*) m

 Come back later, we're
 still working on this one...


(2009-07-23)   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
xm
y,m   =   m y   =     y
vinculum
 xm
 


G s
nm
(2009-07-30)   Christoffel Symbols
Coordinates of the partial derivatives of the basis vectors.

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

In Newtonian mechanics, similar considerations in a "moving" frame of reference entail the introduction of a  rotation vector  and various forces that are 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 . m ê n

Since   ê s . ê n  =  dsn   is a constant, its derivatives vanish and we have:

- G s
nm
    =    ê n . m ê 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)


Ñ  =   ê m Ñm 
(2009-07-29)   Absolute nabla operator   Ñ
Introducing the covariant derivatives  Ñm

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...

tensor field  is a function  (usually not a linear one)  which maps a point  m  of spacetime to some  tensor  T  of rank n.  The  linear  function which maps 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  Ñm  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 Ñm T   f   =   dxm m f

Formally,  Ñm  can thus be defined by  dotting  Ñ  into  êm

Ñm   =    êm . 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:

Ñm ( V )   º   m ( Vn ên )  =  Vn   ên   +    Vnm ên
vinculum
xm
So, ê m m ( Vn ên ) = Vn  ê m ên   +    (ê m m ên ) Vn
vinculum
 xm
 = Vn  ê m ên    +    (ê m m ês ) Vs
vinculum
 xm

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

Ñ ê s     =       - G s
nm
  ê m ê n
m ê 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
Ñm Vn    º    Vn;m    =       Vn    -   G s
nm
 Vs
vinculum
 xm

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

T ab;m     =       Tab    -   G s
ma
 Tsb    -   G s
mb
 Tas
vinculum
xm
 
U abg;m     =       Uabg    -   G s
ma
 Usbg    -   G s
mb
 Uasg    -   G s
mg
 Uabs
vinculum
 xm

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

Ñm V n    º    V n;m    =      V n    +   G n
sm
 V s
vinculum
 xm

Let's just give one example with  mixed  indices  (one upstairs, two downstairs):

U abg;m     =       Uabg    -   G s
am
 Usbg    +   G b
sm
 Uasg    -   G s
gm
 Uabs
vinculum
xm

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


Ñ  =   êmÑ m
(2009-08-03)   Contravariant derivatives   Ñ m
Rare  differentiation along covariant coordinates.

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

Ñ   =   êmÑ m   =   ê m Ñm

Indeed, by dotting everything into  ên  we obtain:

ên . Ñ   =   Ñ n   =   gmn Ñm

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 ( Vn ên )  =  Vn   ên   +    Vn m ên
vinculum
 xm
So, êm m ( Vn ên ) = Vn  êm ên    +    (êm m ên ) Vn
vinculum
 xm
 = Vn  êm ên    +    ( Ñ ês ) Vs
vinculum
xm

The last term is exactly what we found under 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:

Ñ ê s    =     - G s
nm
  ê m ê n     =     - G s
nl
  g lm êm 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 Vn    º    Vn;m    =      Vn    -   G   sm
n
 Vs
vinculum
 xm
Ñ m V n    º    V n;m    =      V n    +   G   nm
s
 V s
vinculum
 xm

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.


(2009-10-21)   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 different reference frames  (K and K)  are related by equations which involve 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 smn   =     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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(2009-10-15)   Einstein's Equivalence Principle
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 smn   =     ½  (  G s
mn
  -   G s
nm
 )
Although, as argued below, Einstein's equivalence principle demands a vanishing torsion  Q,  this is not an absolute requirement.  In fact, "gravity with torsion" is very much needed to account for intrinsic spin as an asymmetric source of gravitation  (besides the ordinary symmetrical stress tensor).  This is the basis of Einstein-Cartan gravity (1922) advocated by Elie Cartan, Dennis Sciama, Tom Kibble, Richard J. Petti, etc.

The so-called  principle of equivalence  postulated in Einstein's  general theory of relativity  implies that spacetime is torsion-free because 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:

Gsmn   =    gsr 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).


(2009-10-23)   Spacetime torsion with 4 independent components
What if the torsion  Qabg 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 smn

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 symmetric part 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   =   - Qsim g sj  -  Qsjm g is   =   - Qjim  -  Qijm   =   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


(2009-08-15)   Levi-Civita symbols:   eij ,   eijk ,   eijkl ,   eijklm ,   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-vectorHodge 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)


 Gregorio Ricci-Curbastro 
 1853-1925
G. Ricci-Curbastro

(2009-07-28)   Ricci's Theorem   (Ricci, 1884)
The covariant derivatives of the  metric tensor  vanish.

Ricci's theorem means that covariant differentiation commutes with the raising or lowering of indices.  This result is dubbed  metric compatibility  and can be construed as the  fundamental theorem of tensor calculusRicci established it in 1884.

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 "suface" 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


(2009-07-31)   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:

[ Ñm , Ñn ] V a  =  Ñm nV a )  -  Ñn mV a )
 = R abmn V b

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

The Riemann-Christoffel curvature tensor :
R abmn     =     ( m G a
bn
  +  G a
ml
  G l
bn
 )   -  ( ¶n G 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 lmln

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


(2009-08-08)   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 mbnl;m   =   0

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

R nn;l  -  R nl;n  +  R mnnl;m   =   0

The  Ricci scalar  R = R nn  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 ml;m   =   0

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

Definition of Einstein's Tensor :
Gmn   =   Rmn - ½ gmn 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  gG  and  T.  The third of those is the  stress tensor  T,  discussed next, whose lack of divergence expresses the conservation of energy and momentum.

Peter Dunsby :   Bianchi identities   |   Einstein tensor


(2009-07-31)   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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(2005-08-22)   Einstein's Field Equations   (Einstein, 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
( Rmn - ½ gmn R )  +  L gmn   =     8 p G   Tmn
vinculum
c 4

The symbols in this relation have the following meanings :

  • gmn   is the  metric tensor  (describing the gravitational potential).
  • Rmn   is the second-rank curvature tensor  (the  Ricci tensor ).
  • R  =  gmn Rmn  =  Rmm   is the  scalar curvature  (or  Ricci's scalar ).
  • Tmn   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 infamous  cosmological constant  (once thought to be zero). 
  • Rmn - ½ gmn R   =   Gmn   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 intrinsinc curvatures are homogeneous to the reciprocal of a surface area  (so is the cosmological constant).

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Wikipedia :   Einstein field equations   |   Einstein-Hilbert lagrangian (Hilbert, 1915)


(2009-07-07)   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 xs   +   G s
mn
  dxm 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 ellapsed 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)   dxm 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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(2009-08-01)   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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Orbits in Strongly Curved Spacetime  by  John Walker
Motion of the Perihelion of Mercury  by  Albert Einstein (1920)


   Karl Schwarzschild 
 1873-1916
Karl Schwarzschild
(2009-08-01)   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).  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)


(2013-12-14)   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)


meglovessims (Yahoo! 2007-08-11)   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).


(2005-08-21)   Unruh Radiation, Unruh Temperature, Unruh Effect
An accelerated observer experiences a  heat bath  of photons.

In 1976, Bill Unruh  (of the University of British Columbia)  showed that an observer submitted to an acceleration  g  (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).


(2005-07-16)   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
An   =   ( -f/c, Ax, Ay, Az )   =   ( -f/c, A )

Covariant Faraday Tensor   F  =  - Rot A
Fmn     =     An,m - Am,n     =     An;m - Am;n

bracket
F00F01F02F03
F10F11F12F13
F20F21F22F23
F30F31F32F33
bracket     =     bracket
0-Ex /c-Ey /c-Ez /c
 Ex /c0Bz-By
 Ey /c-Bz0Bx
 Ez /cBy-Bx0
bracket
bracket   bracket bracket   bracket
bracket bracket bracket bracket

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

bracket
F00F01F02F03
F10F11F12F13
F20F21F22F23
F30F31F32F33
bracket     =     bracket
0 Ex /c Ey /c Ez /c
-Ex /c0Bz-By
-Ey /c-Bz0Bx
-Ez /cBy-Bx0
bracket
bracket   bracket bracket   bracket
bracket bracket bracket bracket

Therefore,   Fmn Fmn   =   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  eo  (  E 2  -  c2 B 2  )   =   - Fmn Fmn / 4mo


(2009-08-05)   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 dimmensions...  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  (denoted by a fifth index equal to 4 ) to the usual 4D spacetime considered so far.  We keep the usual symbols for 4D quantities and primed symbols for their 5D counterparts.  Greek indices run from 0 to 3 and  latin  indices run from 0 to 4.

We  assume  the following relations  (with  g'mn = 0 ) :

g'mn   =   g mn  +  A m A n             g'm4   =   g'4m   =   A m             g'44   =   1

 Come back later, we're
 still working on this one...

Kaluza-Klein theory  by  Christopher N. Pope
Wikipedia :   Kaluza-Klein theory


(2007-08-09)   Harvard Tower Experiment
A delicate demonstration of the gravitational redshift.

 Come back later, we're
 still working on this one...


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

 Come back later, we're
 still working on this one...

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


(2012-12-06)   Warp Drive   (Zefram Cochrane, 2063)
Contract space in front of you and expand space behind you...

 Come back later, we're
 still working on this one...

Discovery Channel :   Can We Travel Faster Than Light?

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