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Gravitational Waves

The Einstein Field Equations are:
R_{μν}  (1/2)g_{μν}R = 8πGT_{μν}
R_{μν} represents the part of curvature that derives from the
presence of matter (i.e. that part due to the stress energy
tensor T_{μν}). The remaining components of the Riemann tensor
(the WEYL tensor) represents the part of the gravitational
field which can propagate as a gravitational wave through a
region of spacetime containing no matter or nongravitational
fields. In other words, these are the parts of the curvature
derived from the dynamics of the gravitational field itself
independent of what matter the spacetime contains. An empty
spacetime containing only gravitational radiation will
satisfy R_{μν} = 0 but will also have R_{μνσ}^{λ} ≠ 0. Regions of
spacetime in which R_{μν} = 0 are referred to as being RICCI
FLAT.
Einstein predicted the existence of gravitational waves. We
start with the field equations in the absence of any mass.
This is a good approximation if there is a large distance
between the source and the observer. Therefore,
R_{μν}  (1/2)g_{μν}R = 0
Therefore,
g^{μν}R_{μν}  (1/2)g^{μν}g_{μν}R = 0
Or,
R  (1/2)δ^{μ}_{μ}R = 0
Now δ^{μ}_{μ} = 4. Therefore,
R  2R = 0
This implies that R = 0. Thus, we can write:
R_{μν} = 0
Note: R is also the trace of the Ricci tensor.
Now consider empty flat space so that g_{μν} = η_{μν}. Consider a
small perturbation is spacetime due to some kind of cosmic
'event'. The metric tensor can be written as:
g^{μν} = η^{μν}  h^{μν}(x,t)
Where h_{μν}(x,t) represents the perturbation.
The Ricci tensor is comprised of the first derivatives and
products of the Christoffel symbols. Thus, schematicaly and
ignoring all indeces in the interest of brevity:
R ~ ∂Γ + ΓΓ
In turn, the Christoffel symbols consist of the metric tensor
and its derivatives. Again, schematically:
Γ ~ (1/2)g^{1}∂g
Where g^{1} is the inverse metric tensor (g^{μν}). Therefore,
Γ ~ (η  h)∂h
If h is very small then the product h∂h can be ignored.
~ η∂h
~ ∂h
Therefore, the Ricci tensor can be written:
R = ∂^{2}h + ∂h∂h
Again, if is small then ∂h∂h > 0. Therefore,
R = ∂^{2}h
This is not just one equation. There is a separate equation
for each μ and ν. Thus,
R_{μν} = [∂^{2}h]_{μν} = 0
[∂^{2}h]_{μν} = 0 has the form of the wave equation:
[∂^{2}h/∂t^{2}  ∂^{2}h/∂x^{2}]_{μν} = 0 ... 1.
NOTE: In the case where the source is close by, the above
equation becomes:
[∂^{2}h]_{μν} = kT'_{μν} where k is a constant.
Where T'_{μν} represents the stress–energy tensor plus quadratic
terms involving h_{μν}. h is involved because the wave field has
an effect on T_{μν}. While the equations are valid everywhere
they are difficulty to solve analytically because of the
complicated source term. Under certain circumstances (flat
space assumption) the quadratic terms involving h can be
ignored making the solutions easier. However, for most
purposes, the zero source solutions are perfectly adequate.
Solutions to 1.0 are of the form:
h_{μν}(t,z) = h^{0}_{μν}sin(k(t  z))
The waves are transverse meaning that waves can only have
components in the plane perpendicular to the propagation of
the wave.
y
 \
  \
  \
 \ z
\  
\ \ 
x \ 
\
Therefore, the time and z components of h are 0.
 
 0 0_{ } 0_{ } 0 
 0 h_{xx} h_{xy} 0 
 0 h_{yx} h_{yy} 0 
 0 0_{ } 0_{ } 0 
 
Now, h is a symmetric tensor so h_{xy} = h_{yx}. Furthermore, there
is another constraint that h_{xx} = h_{yy} (The trace of h_{μν} = 0).
Therefore, we arrive at 3 equations.
h_{xx}(t,z) = h^{0}_{xx}sin(k(t  z))
so that,
g_{xx}(t,z) = 1 + h_{xx}
This corresponds to coordinates that are stretched and contracted
in the x direction. The contraction occurs with the change in sign
of sin(k(t  z).
Similarly, there is stretching and contracting in the y direction:
h_{yy}(t,z) = h^{0}_{yy}sin(k(t  z))
so that,
g_{yy}(t,z) = 1 + h_{yy}
So there is stretching and contracting in the y direction:
Thus, in the xy plane, this looks like.
y
^
:
^

<....< >...> x

v
:
v
Finally, there is a third solution:
h_{xy}(t,z) = h^{0}_{xy}sin(k(t  z))
so that,
g_{xy}(t,z) = 1 + h_{xy}
This corresponds to simultaneous stretching and contracting along
the diagonal direction.
In summary, if a binarypulsar is being observed, the gravitational
waves produced will be observed in the plane perpendicular to the
line of sight.
Wave Origins

Gravitational waves are radiated by objects whose motion
involves asymmetric accelerations. Two objects spinning about a
central axis will not produce gravitational waves but if they
tumble end over end, as in the case of two objects orbiting
each other, they will radiate gravitational waves. The heavier
the masses , and the faster they tumbles, the greater the
gravitational radiation emitted.
In summary, the the following will/will not emit gravitational
waves:
 Two objects orbiting each other.
 A supernova if the explosion asymmetric.
 An isolated nonspinning solid object moving at a constant
velocity will not radiate (conservation of linear momentum)
but one that is accelerating will.
 A spinning disk will not radiate (conservation of angular
momentum).
 A spherically pulsating spherical star will not radiate.
Orbiting Bodies

Gravitational waves carry energy away from their sources and, in
the case of orbiting bodies, this is associated with a decrease
in the orbit. For a simple system of two masses, the radiated
power is given by:
P = dE/dt = 32G^{4}(m_{1}m_{2})^{2}(m_{1} + m_{2})/5c^{5}r^{5}
The amount of radiation emitted corresponds to the loss of
angular momentum of the orbiting body.
In the case of the Earth orbiting the Sun the power emitted is
about 200 watts!
The emission of radiation first circularizes their orbits and
then gradually shrinks their radius increasing the orbital
frequency. The rate of decrease in the orbital radius is
given by:
dr/dt = 64G^{3}(m_{1}m_{2})(m_{1} + m_{2})/5c^{5}r^{3}
In the case of the Earth orbiting the Sun, the Earth's orbit
shrinks by 1.1 x 10^{20} meter per second. At this rate it
will take about 10^{13} times more than the current age of the
Universe to spiral onto the Sun!
Verification

On 2/11/2016 the first experimental evidence of gravitational
waves was announced to the world, thus proving Einstein's
original assertion. The waves were detected by the Laser
Interferometer Gravitational Wave Observatory (LIGO) that
consists of widely separated mirrors that are suspended like
pendulums in a clock. The apparatus is designed such that
the tidal forces associated with gravitational waves causes the
distance between the mirrors to change. The change in distance
between the mirrors is detected using the technique of laser
interferometry.
The discovery of gravitational waves promises to have a
profound impact on our understanding of how the Universe
works.