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General theory of relativity

Coordinate transformation   |   Field equation  |   Gravitational waves  |   Differential Equation

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GRAVITATIONAL WAVES

Gravitational waves are ripples in spacetime, that propagates at the speed of light. Such ripples are the properties of spacetime itself. It propagates at the speed of light.
metric tensor

The proportionality of gravitational and inertial mass, and the " constant of gravitation " which connects them, are conceptions belonging to the approximate Newtonian scheme, and therefore presuppose that the gravitational fields are so weak that the equations can be treated as linear. For more intense fields the Newtonian terminology becomes ambiguous, and it is idle to inquire whether the constant of gravitation really remains constant when the mass is enormously great. Accordingly we here discuss only the limiting case of very weak fields. The metric tensor can be decomposed into two component: one is the usual Minkoswki metric and other is small deviation (h) in the first order. We, can, in this way liniarize gravity which is a non-linear theory according to GR. The wave equation is expressed by De-Alembert () operator.


metric tensor---0

The perturbed metric h is itself a function of coordinates but the condition is that |h| << 1 so that higher order terms can be neglected. This is the method of perturbation which can be used to solve specific problem by approximating the solution by finding exact solution of related , simpler problem. where δ_uv, represents Galilean values, and h_uv„ will be a small quantity of the first order whose square is neglected. The derivatives of the g^v will be small quantities of the first order. We have, correct to the first order
metric tensor-----1
by B_σαμ&neu; =
metric tensor
We shall try to satisfy this by breaking it up into two equations
metric tensor--2
The second equation becomes, correct to the first order,
metric tensor
This is satisfied if
metric tensor --3

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The other equation can be written as
metric tensor
showing that G_uv is a small quantity of the first order. Hence
metric tensor
This "equation of wave-motion" can be integrated. Since we are dealing with small quantities of the first order, the effect of the deviations from Galilean geometry will only affect the results to the second order ; accordingly the well-known solution* may be used, viz.
metric tensor
the integral being taken over each element of space-volume dV at a coordinate distance r from the point considered and at a time t at r', i.e. at a time such that waves propagated from clV with unit velocity can reach the point at the time considered.
If we calculate from the value of the
metric tensor
the operator d/dx_a indicates a displacement in space and time of the point considered, involving a change of r . We may, however, keep r' constant on the right-hand side and displace to the same extent the element dV where (Tα_μ)' is calculated. Thus
metric tensor
d(Tab)/dx_a is of the second order of small quantities, so that to our approximation (3) is satisfied.
The result is that
gravitational waves einstein
where De'alembert's operator is used.
gravitational waves einstein
satisfies the gravitational equations correctly to the first order, because both the equations into which we have divided (1) then become satisfied. Of course there may be other solutions of (1), which do not satisfy both equations of 2 separately.
For a static field the avove equation reduces to
metric tensor
Also for matter at rest T= T_44 = p (the inertial density) and the other components of T_uv vanish ; thus
graviational wave equation
For a single particle the solution of this equation is well known to be h_11, h_22, h_33, h_44 = -2m/r
Hence by (0) the complete expression for the interval is
graviational wave equation

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In quantum field theory perturbation method is also useful in many cases when the specific problem is harder to solve.

metric tensor

This is the the gravitational wave equation in non-homogeneous form where a source term (T) is present on the right hand side. It has the effect on the medium which carry gravitational wave. The above equation can be derived from field equation. It is the wave equation in spactime , which encodes the properties of undulation of spacetime. The phenomena is the same as the propagation of electromagnetic field. In case of the latter the charged particle accelerates and in the former case heavier objects collides or accelerates. As an example when two neutron stars collides gravitational waves are created. But how can it be detected when spacetime is itself changing?


progation of gravitational wave
gravitational wave equation

Gravitational waves can penetrate regions of space that electromagnetic waves cannot. They are able to allow the observation of the merger of black holes and possibly other exotic objects in the distant Universe. Such systems cannot be observed with more traditional means such as optical telescopes or radio telescopes, and so gravitational wave astronomy gives new insights into the working of the Universe. In particular, gravitational waves could be of interest to cosmologists as they offer a possible way of observing the very early Universe. This is not possible with conventional astronomy, since before recombination the Universe was opaque to electromagnetic radiation. Precise measurements of gravitational waves will also allow scientists to test more thoroughly the general theory of relativity.

Jouney into the realm of physics

In 1900, the British physicist Lord Kelvin is said to have pronounced: "There is nothing new to be discovered in physics now. All that remains is more and more precise measurement." Within three decades, quantum mechanics and Einstein's theory of relativity had revolutionized the field. Today, no physicist would dare assert that our physical knowledge of the universe is near completion. To the contrary, each new discovery seems to unlock a Pandora's box of even bigger, even deeper physics questions. These are our picks for the most profound open questions of all. Inside you’ll learn about parallel universes, why time seems to move in one direction only, and why we don’t understand chaos.


What is dark energy?

No matter how astrophysicists crunch the numbers, the universe simply doesn't add up. Even though gravity is pulling inward on space-time — the "fabric" of the cosmos — it keeps expanding outward faster and faster. To account for this, astrophysicists have proposed an invisible agent that counteracts gravity by pushing space-time apart. They call it dark energy. In the most widely accepted model of dark energy, it is a "cosmological constant": an inherent property of space itself, which has "negative pressure" driving space apart. As space expands, more space is created, and with it, more dark energy. Based on the observed rate of expansion, scientists know that the sum of all the dark energy must make up more than 70 percent of the total contents of the universe. But no one knows how to look for it. The best researchers have been able to do in recent years is narrow in a bit on where dark energy might be hiding, which was the topic of a study released in August 2015.

More on De'alembert operator


progation of gravitational wave

De-alembert operator

De-alembert operator
There is a principle called De-alembert's principle . It can be stated as below :

De-alembert principle

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