General 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 remarked that
the force exerted by gravity on a object
[which we call the weight of that object]
is strictly proportional to its inertial mass, just like any
of 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 fictitious 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 corresponding 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 generalizieren.
Edward "Ned"
van Vleck (1916)
(whose son, John Hasbrouck van Vleck, earned a
Nobel prize in 1977)
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).
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 rotation 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
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 |
|
|
1 -
(W´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
Sagnac Time Lag
(observer tied to the loop)
| Dt'
= |
4 W . S |
|
| 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 in the material
the solid is built from.
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 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 !)
(2003-11-03)
Covariant Derivatives
(2009-07-07)
Motion of a Free-Falling Particle
Proper time is extremal along the spacetime path of freefall.
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.)
which do apply here.
As "time" is just one of the spacetime coordinates, another arbitrary parameter
l is used to describe a spacetime path
Q(l) of fixed extremities
along which the Lagrangian integrand is simply proportional to the interval of
proper time :
[ g mn(Q)
dx m
dx n ] ½
This is a straight variational problem with a
Lagrangian L(Q,V) proportional to
[ g mn(Q)
v m
v n ] ½
(2005-08-22)
Einstein's Field Equations
Einstein's Law of the Gravitational Field
| ( Rmn -
½ gmn R )
+
L gmn
=
| 8 p G |
Tmn |
|
| c 2 |
|
The symbols in this expression have the following meanings :
- gmn
is the tensor of the gravitational potential.
- Rmn
is Riemann's second-rank curvature tensor
(the Ricci tensor ).
- R = gmn Rmn
is the scalar curvature (or Ricci scalar ).
- Tmn
is the stress-energy tensor (expressing the presence of energy).
- G is Newton's constant of gravity (about
6.67428(67) ´ 10-11 SI ).
- L is the controversial
cosmological constant, which could be zero.
- Rmn -
½ gmn R
= Gmn is the Einstein tensor.
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 r
meglovessims (Yahoo!
2007-08-11)
What is mass ?
Is mass a property of matter?
Relativistic 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 those two approaches end up defining exactly
the same thing is the equivalence principle,
the basic tenet of Einstein's General Theory of 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 and Unruh Temperature
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, equivalently, 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 convariant 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
|
|
|
| F00 | F01 | F02 | F03 |
| F10 | F11 | F12 | F13 |
| F20 | F21 | F22 | F23 |
| F30 | F31 | F32 | F33 |
|
|
= |
|
| 0 | -Ex /c | -Ey /c | -Ez /c |
| Ex /c | 0 | Bz | -By |
| Ey /c | -Bz | 0 | Bx |
| Ez /c | By | -Bx | 0 |
|
|
| |
| |
| |
| |
|
|
|
|
(2007-08-09)
Harvard Tower Experiment
A delicate demonstration of the
gravitational
redshift.
(2009-04-10)
Shapiro Delay
(Irwin I. Shapiro, 1964)
Gravitational time dilation causes apparent delays in radar signals.
Wikipedia :
Shapiro Delay
|
