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
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).
Einstein's Theory of General Relativity by
Leonard Susskind
(Fall 2008, Stanford University continuing education)
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iTunesU General Relativity
(Susskind, Fall 2012)
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General Theory of Relativity
Introduction :
In classical mechanics, it's often convenient to describe motion in
noninertial frames of reference (e.g., a rotating
coordinate system). In such a system, the laws of mechanics won't hold
unless we use particular expressions for the derivative of a vector
(the acceleration and rotation vector of the frame
of reference itself are involved).
Alternately, we may apply to any frame of reference the laws of mechanics in the form
they assume in an inertial frame, provided we introduce special
fictitious forces
proportional to the mass of the object
(like the centrifugal force or the Coriolis force).
We could always bypass either approach and analyze the problem with respect to
an inertial coordinate system
(no matter how contrived a construction this inertial system may be).
We did just that in our analysis of the
Coriolis effect in free fall and
the Sagnac effect.
Albert Einstein remarked that
the force exerted by gravity on an object
[which we call the weight of that object]
is strictly proportional to its inertial mass, just like
the aforementioned fictitious forces.
He dubbed this observation the equivalence principle
(i.e., inertial mass and gravitational mass are one and the same)
and drew all the consequences
of putting gravitational forces and inertial forces on the same footing.
The only difference between gravity and an ordinary fictitious force field
is that the former cannot (usually) be reduced to a mere artifact of
coordinate motion. So, with gravity, we no longer have the luxury
of going back at will to an "inertial frame" where physical laws are simpler.
Instead, we're stuck with a system of coordinates corresponding
to whatever the local geometry becomes because of the presence of gravity.
The ensuing mathematical framework is the stage for the General Theory of Relativity.
This stage was not left empty by Einstein, who came up with a compatible description
of how gravity is produced by mass (or, rather, energy).
This ends up relating the curvature of spacetime with the distribution of energy in it.
The result is Einstein's field equations.
The mathematics involved may be intimidating but the basic principles
(stated above) are quite simple. The implications are mindboggling.
Louis Vlemincq (20050725; email)
Observer on a Rotating Disc
Does the HarressSagnac effect
contradict General Relativity ?
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 solidstate
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).
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
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
halfsilvered 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 / c^{2 })
W . S
1 
(W´r)^{2} / c^{2}
Note the bold type indicating vectorial quantities,
defined as follows:
r is the 3D position, relative to an origin on the
axis of rotation.
S is the loop's
vectorial surface,
an axial vector which depends not only on the
conventional orientation of space but also on which direction is chosen as
positive to travel around the loop.
In Euclidean geometry, S may
be defined by a contour integral around the oriented
loop (C+).
S
= ½
ò_{C+}r ´ dr
For a closed loop C+ this defining integral does not depend on the
arbitrary origin chosen for the position vector r.
Anybody encountering this for the first time is encouraged to
work out S 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
Sagnac Time Lag
(observer tied to the loop)
Dt'
= _{ }
4 W . S
c^{ 2}
(20050729)
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 vectorW
which ties the velocities of any pair A and B
of the solid's points, via the following
relation ( W is
an axial vector whose sign depends on space "orientation").
v_{A} 
W ´ A =
v_{B} 
W ´ B
This is only a good approximation to physical reality if any change in the velocity
of a point is somehow made known instantly throughout the
solid so that the relative distance of all pairs of its points can be maintained...
In practice, however, such information can be propagated no faster than the
speed of sound within the solid.
Loosely speaking, a change in rotation which starts at the axis of rotation
will propagate at the slower tranverse speed,
while other changes propagate at a speed intermediary between this speed
(Swaves) and the true speed of sound (Pwaves).
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 !)
(20090725)
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 twodimensional 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,
ê_{1} and ê_{2}
which need not be orthogonal and need not be of unit length...
Any infinitesimal twodimensional vector dV
is a linear combination whose coefficients are said to be
its [contravariant] coordinates in that basis:
dV = dv^{1}ê_{1}
+ dv^{2}ê_{2}
By definition, the covariant coordinates
of dV (endowed with lower indices)
are obtained by dotting dV into
ê_{1} and ê_{2}
respectively:
dV . ê_{1} = dv_{1}
and
dV . ê_{2} = dv_{2}
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,
ê^{1} and ê^{2}
defined by:
Those defining relations can be summarized using the
Kronecker delta symbol,
in a way which remains true in any number of dimensions:
ê_{ i} . ê^{ j} =
d_{i}^{j}
( equal to 1 if i = j and zero otherwise)
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
(18111863, X1829) who held the chair of physics at Polytechnique from 1845 to 1856).
The following relations thus hold:
dV
=
dv_{1}ê^{1}
+ dv_{2}ê^{2}
=
(dV.ê_{1 }) ê^{1}
+ (dV.ê_{2 }) ê^{2}
=
dv^{1}ê_{1}
+ dv^{2}ê_{2}
=
(dV.ê^{1 }) ê_{1}
+ (dV.ê^{2 }) ê_{2}
 dV ^{ 2}
=
dv^{1} dv_{1} + dv^{2} dv_{2}
All this is best expressed by introducing the metric tensor
g_{mn}^{ } (which
defines the dot product and, hence, the notion of length).
g_{mn}
is symmetric ( g_{12} = g_{21 })
because the dot product is commutative
( ê_{1 }. ê_{2} =
ê_{2 }. ê_{1 }).
ê_{1}
=
g_{11}ê^{1} + g_{12}ê^{2}
=
(ê_{1 }. ê_{1 }) ê^{1} +
(ê_{1 }. ê_{2 }) ê^{2}
ê_{2}
=
g_{21}ê^{1} + g_{22}ê^{2}
=
(ê_{2 }. ê_{1 }) ê^{1} +
(ê_{2 }. ê_{2 }) ê^{2}
By definition, the square of the length of dV is the dot product
dV . dV :
The
matrix g^{mn}
that appears in this last expression is the multiplicative inverse
of g_{mn} which
expresses the reciprocal linear relations, namely:
ê^{1}
=
g^{11}ê_{1} + g^{12}ê_{2}
=
(ê^{1 }. ê^{1 }) ê_{1} +
(ê^{1 }. ê^{2 }) ê_{2}
ê^{2}
=
g^{21}ê_{1} + g^{22}ê_{2}
=
(ê^{2 }. ê^{1 }) ê_{1} +
(ê^{2 }. ê^{2 }) ê_{2}
The following relation holds in any number of dimensions:
To draw a nice picture where the circle of unit radius
does look like a circle, we use an
orthonormal grid in which:
ê_{1} = ( 1, 0 )
ê_{2} = ( 1, 2 )
The metric tensor
( g_{ ij} = ê_{i }. ê_{j })
is :
[
g_{ ij }
]
=
1
1
1
5
Its inverse is:^{ }_{ }
[
g^{ ij }
]
= ¼
5
1
1
1
Therefore:
ê^{1}
=
(5 ê_{1} + ê_{2 })/4
=
( 1, ½ )
ê^{2}
=
( ê_{1} + ê_{2 })/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, 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 :
The distinction between "contravectors" and "covectors" is misguided
because the metric itself establishes
a firm onetoone 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:
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
semiRiemannian
manifold.
(20090807)
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'
We use the "same basis" as before:
ê_{1} = ( 1, 0 )
ê_{2} = ( 1, 2 )
The metric tensor
( g_{ ij} = ê_{i }. ê_{j })
is :
[
g_{ ij }
]
=
1
1
1
3
Its inverse is:^{ }_{ }
[
g^{ ij }
]
= ¼
3
1
1
1
Therefore:
ê^{1}
=
(3 ê_{1}  ê_{2 })/4
=
( 1,  ½ )
ê^{2}
=
(  ê_{1}
 ê_{2 })/4
=
( 0,  ½ )
(20090805)
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
onetoone correspondence with each other.
Such a bijective
correspondence is called a metric
and is fully specified by a nondegenerate
quadratic form, denoted by a dotproduct
("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 socalled Lorentzian metric of
fourdimensional 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 onetoone
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 n^{th}rank tensor, or ntensor)
is a linear function that maps a vector to a tensor of rank n1.
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).
(20090729)
Signature of a Quadratic Form
Bases in which a given metric tensor has its simplest expression.
In the previous introductory article, we defined
the metric tensor with respect to a particular basis
in terms of a known ordinary euclidean dot product:
g_{ ij} = ê_{ i} . ê_{ j}
From that metric tensor alone, we
computed reciprocal vectors satisfying:
ê_{ i} . ê^{ j}
=
d_{i}^{ j}
It turns out that this can always be done
if we define ab initio
our "dot product"
(which need not result in a positive definite quadratic form)
by specifying the aforementioned metric tensor
to be any given symmetric matrix (invertible or not).
Furthermore, there are special vector bases where the dot product
so defined has a particularly simple expression, namely:
g_{ ij} = 0
when i differs from j.
g_{ ii} is equal to 0,
1, or +1.
More loosely, we only need a matrix M such that
M g M* is diagonal.
In all such cases, the numbers of negative and positive quantities on
the diagonal are the same and they define what's called the
signature of the metric.
(If there are no zeroes on the diagonal,
the metric is said to be nondegenerate.)
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).
ê_{ i}. ê^{ j}
=
g_{ i}^{j}
= d_{ i}^{j}always
(by definition of the reciprocal vectors).
ê_{ i}. ê^{ }_{j}
=
g^{ }_{ij}
= h_{ ij}
in an orthonormal basis only.
For a nondegenerate metric,
h_{ij} = 0
when i ¹ j whereas
h_{ii} = ± 1.
(20090721)
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:
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 xcoordinates.
The differential of each ycoordinate is thus
a linear combination of the differentials of the four xcoordinates.
By definition, the coefficients of those linear combinations are known
as partial derivatives :
d y^{ m}_{ } =
¶y^{m}
_{ } dx^{0} +
¶y^{m}
_{ } dx^{1} +
¶y^{m}
_{ } dx^{2} +
¶y^{m}
_{ } dx^{3}
¶x^{0}
¶x^{1}
¶x^{2}
¶x^{3}
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 welldefined
"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}_{ } =
¶y^{m}
_{ } d x^{ n}
¶x^{n}
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]
_{ } =
¶y^{m}
_{ } V^{ n }[x]
¶x^{n}
Instead of using different sets of names, we may
underscore whatever relates to the second frame of reference
(vectorial components,
coordinates, differential operators with respect to coordinates, etc.).
The above becomes:
Transformation of Contravariant Coordinates
V^{m}
=
¶_{n}x^{m}
V^{n}
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
Note that the quantity on the lefthand side lacks any "open"
index because we are referring to the mathematical object itself,
as opposed to its coordinates in a particular frame of reference.
We shall henceforth use bold type to denote
an object with components that are not made explicit
by an apparent index (loosely speaking, there are
hidden indices in a bold symbol).
There's also an implication that a given object could be described by
other schemes besides the aforementioned contravariant
linear combinations.
Indeed, one such scheme is the covariant viewpoint which we are about
to describe (both aspects become interchangeable in a
metric space).
Gradients and other covariant coordinates :
Transformation of Covariant Coordinates
V_{m}
=
¶_{n} x^{m}
V_{n}
(20090721)
The metric tensor
g_{mn}
and its inverse g^{mn} Lowering or raising indices.
The spacetime interval (squared) is
g_{mn}
dx^{m}
dx^{n}
(20090725)
Duality A dual
is obtained by switching all indices (and
complex conjugation).
(20090723)
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 viceversa.
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}
=
¶_{m}y
^{ } =
¶^{ }y
¶_{ }x^{m}
y^{,m}
=
¶^{ m }y
_{ } =
¶^{ }y
¶^{ }x_{m}
(20090730) 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 socalled 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}
= d^{s}_{n} 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!]
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
(20090729)
Absolute nabla operator
Ñ
Introducing the covariant derivatives
Ñ_{m}
Covariant derivatives are due to Gregorio
RicciCurbastro (18531925) who invented most of tensor calculus
between 1884 and 1894
(Delle derivazione covariante e contravariante, Padova, 1888).
In 1900,
with his former student
Tullio LeviCivita
(18731941) Ricci published a 75page masterpiece entitled
Méthodes de calcul différentiel absolu et leurs applications.
That treatise unified and extended the pioneering efforts of
Carl Friedrich Gauss (17771855),
Bernhard Riemann (18261866) and
Elwin
Christoffel (18291900).
It was Marcel
Grossmann (18781936) 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 (18801933) had
pointed out (in 1909) the noneuclidean character of geometry in one
particular noninertial frame of reference: The rim of a
spinning circular platform measures less than
p times its diameter !
Equating inertial accelerations and gravitational fields
(his principle of equivalence)
Einstein suspected that gravity might be related to
a local disturbance in the metric features of spacetime...
A tensor field is a function (usually not a linear one)
which maps a point m of spacetime to some
tensorT 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
Ñ_{m} is to the
absolute differentiation of a tensor T
what the partial derivative
¶_{m} is to the
differentiation of a scalar f.
Ñ T =
ê^{ m}
Ñ_{m }T
d f =
dx^{m }
¶_{m}f
Formally,
Ñ_{m}
can thus be defined by dotting
Ñ into
ê_{m}
Ñ_{m} = ^{ }ê_{m}. Ñ
The crucial difficulty is that a tensor of nonzero rank is obtained
by summing every coordinate multiplied into a matching
tensorproduct of basis vectors. The derivative of every such term
is obtained by the product rule.
Only one component is obtained by differentiating the
coordinate itself; all the other components
involve derivatives of the basis vectors.
We'll first restate this in the case of tensors of rank 1 (vectors)
before generalizing again to tensors of any rank.
A vector is really a linear combination of basis vectors which may
well change as the differentiation variable varies.
Therefore, the product rule fully applies
(that's similar to the way
rigid motion brings about a
rotation vector ) and we obtain:
Ñ_{m}^{ }( V )
º
¶_{m}^{ }( V_{n}ê^{n })
=
¶_{ }V_{n}
ê^{n}_{ }
+
^{ }V_{n} ¶_{m}ê^{n}
¶^{ }x^{m}
So,
ê^{ m }
¶_{m}^{ }( V_{n}ê^{n })
=
¶_{ }V_{n}
ê^{ m}ê^{n}_{ }
+
^{ }(ê^{ m } ¶_{m}ê^{n }) V_{n}
¶^{ }x^{m}
=
¶_{ }V_{n}
ê^{ m}ê^{n}_{ }
+
^{ }(ê^{ m } ¶_{m}ê^{s }) V_{s}
¶^{ }x^{m}
Since ¶_{m} and
Ñ_{m} have the same effect on basis vectors,
what appears in the last bracket is actually the nabla operator
Ñ =
ê^{ m}
Ñ_{m} applied to ê^{ s}.
The coordinates of that
are the ^{ }Christoffel symbols introduced above:
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
Ñ_{m}
V_{n} º
V_{n;m}
=
¶_{ }V_{n}
 G
s nm
^{ }V_{s}
¶^{ }x^{m}
The covariant derivative of a tensor of rank n entails a sum of n+1 terms:
T_{ ab;m} =
¶_{ }T_{ab}
 G
s ma
^{ }T_{sb}
 G
s mb
^{ }T_{as}
¶^{ }x^{m}
U_{ abg;m} =
¶_{ }U_{abg}
 G
s ma
^{ }U_{sbg}
 G
s mb
^{ }U_{asg}
 G
s mg
^{ }U_{abs}
¶^{ }x^{m}
If upper indices are used, the coordinates of contravariant derivatives obey
a similar rule with the same symbols, but different summations
and opposite signs:
Ñ_{m}
V^{ n} º
V^{ n}_{;m}
=
¶_{ }V^{ n}
+ G
n sm
_{ }V^{ s}
¶^{ }x^{m}
Let's just give one example with mixed indices
(one upstairs, two downstairs):
U_{ a}^{b}_{g;m} =
¶_{ }U_{a}^{b}_{g}
 G
s am
^{ }U_{s}^{b}_{g}
+ G
b sm
^{ }U_{a}^{s}_{g}
 G
s gm
^{ }U_{a}^{b}_{s}
¶^{ }x^{m}
This simple statement summarizes 256 formulas,
with 13 terms each...
(20090803)
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
=
ê_{m}Ñ^{ m}
=
ê^{ m}
Ñ_{m}
Indeed, by dotting everything into
ê^{n} we obtain:
ê^{n}. Ñ
=
Ñ^{ n}
= g^{mn} Ñ_{m}
That was merely a consistency check:
Since the covariant derivative of a tensor is known to be a tensor,
we are certainly allowed to raise the index which appears downstairs
after a covariant differentiation...
We may also obtain expressions for contravariant derivatives
ab initio :
Ñ^{ m}_{ }( V )
º
¶^{ m}_{ }( V_{n}ê^{n })
=
¶_{ }V_{n}
ê^{n}_{ }
+
^{ }V_{n} ¶^{ m}ê^{n}
¶_{ }x_{m}
So,
ê_{m }¶^{ m }( V_{n}ê^{n })
=
¶_{ }V_{n}
ê_{m}ê^{n}_{ }
+
^{ }(ê_{m }¶^{ m}ê^{n }) V_{n}
¶^{ }x_{m}
=
¶_{ }V_{n}
ê_{m}ê^{n}_{ }
+
^{ }( Ñ
ê^{s }) V_{s}
¶^{ }x_{m}
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 componentwise relations:
Ñ_{ }ê^{ s} =
 G
s nm
ê^{ m}ê^{ n}
=
 G
s nl
g^{ lm}ê_{m}ê^{ n}
We may thus introduce the following notsocommon notation to simplify the
explicit expressions for contravariant derivatives given below:
G
^{ }
sm n
=
G
s nl
g^{ lm}
Contravariant derivatives of a vector
Ñ^{ m}
V_{n} º
V_{n}^{;m}
=
¶_{ }V_{n}
 G
^{ }
sm n
^{ }V_{s}
¶_{ }x_{m}
Ñ^{ m}
V^{ n} º
V^{ n;m}
=
¶_{ }V^{ n}
+ G
^{ }
nm s
_{ }V^{ s}
¶_{ }x_{m}
This is just for completeness...
Those explicit formulas for contravariant derivatives
are rarely used, if ever.
They can be generalized to tensors of higher ranks by using the same patterns as
covariant derivatives.
(20091021) 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^{ s}_{mn} =
½ (
G
s mn

G
s nm
)
Elie Cartan first described spaces with nonzero
torsion in 1922 (Einstein's equivalence principle
implies Q = 0 but this is not a logical
requirement).
When zero torsion is not assumed, General Relativity becomes what's known
as EinsteinCartan theory (Einstein himself thought that Q
might describe electromagnetism; it doesn't).
Spacetime torsion is needed if intrinsic pointlike
spin is allowed.
Formally,
(20091015) Einstein's Equivalence Principle
A postulate implying the symmetry of Christoffel symbols.
The Christoffel symbols
do not form a tensor, but the following
quantity is a proper antisymmetric tensor called
Cartan torsion (or Cartan tensor ) :
Q^{ s}_{mn} =
½ (
G
s mn

G
s nm
)
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 EinsteinCartan gravity (1922) advocated by
Elie Cartan,
Dennis Sciama,
Tom Kibble,
Richard J. Petti, etc.
The socalled principle of equivalence
postulated in Einstein's
general theory of relativity implies that spacetime is torsionfree
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 torsionfree spacetime is
metriccompatible, then
the Christoffel symbols are functions of the metric coefficients
and their first derivatives:
The advertised result is obtained by multiplying both sides into
½ g^{ sl}
Note that,
conversely, the above formula only holds in the torsionfree
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).
(20091023) Spacetime torsion with 4 independent components
What if the torsion Q_{abg}
is a totally antisymmetric tensor...
With nonzero torsion
in a metriccompatible geometry,
the final summation in the above proof yields the
following equation:
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:
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 torsionfree case.
The remaining antisymmetric part boils down to:
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...
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
kvector.
Hodge duality is a linear bijection between
kvectors and (nk)vectors.
W. V. D. Hodge (19031975).
(20090728)
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 calculus.
Ricci 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
paralleltransported vectors remain constant.
The situation is simpler than it sounds.
One elementary way to visualize it is to consider the special case of
a twodimensional curved surface in Euclidean threedimensional
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 higherdimensional
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 higherdimensional 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.
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.
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_{ mn} =
R^{ l}_{mln}
Because of the symmetries of the Riemann tensor,
the Ricci tensor is
(up to a sign change) the only
nonvanishing contraction of the Riemann tensor.
Luigi Bianchi (18561928)
rediscovered the identities named after him in 1902.
They had first been discovered in the early 1880's by his former classmate
Gregorio RicciCurbastro
who had forgotten all about it
(according to Tullio LeviCivita,
the main collaborator and only former doctoral student of Ricci's).
The Bianchi identity_{ }:
R_{ abmn;l} +
R_{ ablm;n} +
R_{ abnl;m} = 0
A contracted version holds for the
Ricci tensor (HINT:
multiply by g^{ am }).
Contracted Bianchi identity_{ }:
R_{ bn;l}

R_{ bl;n}
+
R^{ m}_{bnl;m} = 0
By contracting this with respect to the indices
b and
n, we obtain:
R^{ n}_{n;l}

R^{ n}_{l;n}
+
R^{ mn}_{nl;m} = 0
The Ricci scalar R = R^{ n}_{n}
appears in the first term. The second and third terms happen to be equal.
So, the whole relation boils down to:
R_{ ;l}

2 R^{ m}_{l;m} = 0
That key relation establishes that the following tensor, introduced by Einstein,
has a vanishing divergence
(i.e., G^{m}_{n;m} = 0 ).
Definition of Einstein's Tensor_{ }:
G_{mn} =
R_{mn} 
½ g_{mn} R
Besides the metric tensorg
itself, the Einstein tensorG
turns out to be the only divergencefree secondrank 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
secondrank tensors g, G and T.
The third of those is the stress tensorT, discussed
next, whose lack of divergence expresses the conservation
of energy and momentum.
(20090731)
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 4dimensional 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 stressenergy or
energymomentumstress.)
(20050822)
Einstein's Field Equations
(Einstein, 1915)
Matter tells space how to curve,
and space tells matter how to move.
John Archibald Wheeler ^{ }
(19112008)
Einstein's Law of the Gravitational Field
( R_{mn} 
½ g_{mn} R )
+
L g_{mn}
=
8 p G
T_{mn}
c^{ 4}
The symbols in this relation have the following meanings_{ }:
g_{mn}
is the metric tensor (describing the gravitational potential).
R_{mn}
is the secondrank curvature tensor (the Ricci tensor ).
R = g^{mn} R_{mn}
= R^{m}_{m}
is the scalar curvature (or Ricci's scalar ).
T_{mn}
is the stressenergy 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)._{ }
R_{mn} 
½ g_{mn} R
= G_{mn} is
Einstein's tensor.
The elements of the stress tensor T
are in units of
energy density or pressure (same thing; a pascal
is a joule per cubic meter or a newton per square meter).
If the coordinates are all in distance units (they need not be) then
the metric tensor is dimensionless and the intrinsinc curvatures are
homogeneous to the reciprocal of a surface area (so is the cosmological
constant).
(20090707)
Motion of a FreeFalling 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 ^{ }
(19112008)
Along a geodesic, the secondorder variation of position vanishes:
d^{ 2 }x^{s}_{ } + G
s mn
_{ }
dx^{m}
dx^{n}
= 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.
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 :
This is a straight variational problem with a
Lagrangian L(Q,V) proportional to
[ g_{ mn}(Q)
v^{ m}
v^{ n} ]^{ ½}
(20090801) 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:
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 ).
(20090801) The Schwarzschild Metric (1915)
An exact solution due to
Karl Schwarzschild
(18731916)
In 1923,
George David Birkhoff (18841944)
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
(18881922) but it wasn't promoted because of Jebsen's battle with turberculosis (in spite of
C.W. Oseen's efforts).
This represents the relativistic gravitational field around a (structureless)
point of mass M if we let the socalled
Schwarzschild radius be:
a = 2 G M / c^{2}
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:
EddingtonFinkelstein coordinates
KruskalSzekeres coordinates
Achilles & the Tortoise (Zeno).
PainlevéGullstrand coordinates (PG) :
This proposal is now of historical interest only.
It was originally mistaken as an alternative sphericallysymmetric 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 crossterm 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.
meglovessims (Yahoo!
20070811)
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 socalled 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/c^{2}
(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).
(20050821)
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 / (4p^{2}c)
[ 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 particleantiparticle creation.
Unruh radiation is thus similar to the betterknown 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).
(20050716)
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 4dimensional rotational of the covariant potential A:
In flat space (no gravity) the doublycontravariant coordinates
of F are:
F^{00}
F^{01}
F^{02}
F^{03}
F^{10}
F^{11}
F^{12}
F^{13}
F^{20}
F^{21}
F^{22}
F^{23}
F^{30}
F^{31}
F^{32}
F^{33}
_{ } = _{ }
0
E_{x }/c
E_{y }/c
E_{z }/c
E_{x }/c
0
B_{z}
B_{y}
E_{y }/c
B_{z}
0
B_{x}
E_{z }/c
B_{y}
B_{x}
0
Therefore,
F_{mn}
F^{mn}
=
2 (  E^{2}/c^{2}
+ B^{2 })
which is proportional to the Lagrangian density compatible with
the Hamiltonian energy density derived from the
Poynting theorem, namely:
(20090805)
KaluzaKlein Theory
Using a fifth spacetime dimension to explain electromagnetism.
The theory formulated by
Theodor Kaluza (18851954)
in 1919 and refined by
Oskar Klein (18941977) 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 5dimensional KaluzaKlein 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 5dimensional 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 ¶_{4 }g'_{mn} = 0 ) :
g'_{mn} =
g_{ mn} +
A_{ m }A_{ n}
g'_{m4} =
g'_{4m} =
A_{ m }
g'_{44} = 1