What are Einstein's Gravitational Waves and why do they matter?

To be very specific it's one of those pioneering discoveries that has the capability to change the way we see the world. It's one of the predictions of Albert Einstein in support of General Theory of Relativity. Others include Worm whole,White whole,gravitational lensing.

Since the birth of "science", human minds were inquisitive about the history of the universe and how the life has taken shape over the time. Theories involving electromagnetic radiation such as visible light, x-rays, radio waves, microwaves and sub atomic particles fought each other to be in the first position of being the true reason for our existence however Einstein believed that whatever happened in past actually left "footprints" in some form.

The General theory of Relativity (which was formulated exactly a hundred years from now) was undoubtedly one of the boldest and most pioneering contributions made by Einstein. He went beyond Newton's Law of Universal gravitation and conceptualised gravity as not merely a force, but as a geometric property of the space-time model and a consequence of the inherent curvature of space-time. According to GTR, space-time isn't flat at all but curved in a highly complex manner. The underlying mathematics of General Relativity is Riemannian geometry, which is a generalised form of Euclidean geometry that deals with Non Euclidean space, Though I don't have in-depth knowledge about it and whatever I explained is in terms of popular science level.

Now the curvature of space time is directly related to the energy or momentum of the present matter or radiation expressed in Einstein’s set of ten field equations. The curvature is caused by the presence of mass and changes proportionally with mass volume ratio i.e. density. As objects with mass move around in space- time,the curvature changes to reflect the changed locations of those objects. In some cases, massive accelerating objects disrupt space and time in such way that the “information” passes outwards in the form of waves and with same speed as that of electromagnetic waves. These wave are known as gravitational waves. Depending on the celestial event, collision of neutron stars or dwarf stars, changes the nature of gravitational waves.

As we know LIGO was in charge of detecting those gravitational waves and till yesterday it was still a mystery. Some days back Laurence Krauss tweeted that LIGO has already found enough signs and would officially announce the news. Well here it is!! Albert Einstein really could predict the future.

Again I would like to say, I am not an expert and hardly have any idea about GTR mathematics. If an expert is here, please care to contribute.

Kaustubh Katdareand can explain in easy to understand words; I'm all ears.

The gravity of the question weighs upon me. I am no expert.
Einstein proposed that gravitational fields can cause space to curve. This has been amply verified. However, gravitation is a weak force, which makes it difficult to verify experimentally some of the consequences.
The special theory limits the speed of information transfer to the speed of light. Einstein's solution of the gravitational equations showed that when masses accelerate they generate time dependent gravitational fields. These propagate through space as warpages of space-time called gravitational waves. The speed of propagation is limited to the speed of light.

Gravitation being a weak force, these waves are difficult to detect. Gravitation is directly proportional to the accelerating mass. Black holes have very large masses. When two black holes are on a collision course things happen, which is what is in the news. LIGO (Laser Interferometer Gravitaional-Wave Observatory) has detected such an event.
As to 'why do they matter?', George Mallory's retort on being asked why he wants to climb the Everest:'"because it's there" about sums it up.
This article (not overly technical) in Physics Today is worth the time:
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If anyone lives in Pune, you can attend the lecture on GW at IUCAA,Pune. Lecture starts from 7pm today and it's an open to all lecture.

From what I know, the speed at which GW travel is not yet pinned down to c (careful, that not the same as the "speed of light"), although one of the experts claim it is a trivial formality to complete. GW are disturbances in space-time that are created due to mass/energy interactions and that they travel (akin to EM waves). They are extremely weak and hence were not detected for long (predicted yes, detected no). From what I read, they increased the sensitivity of LIGO and (viola!) so soon after, we have the finding! Imagine the ripples in a pond when you throw a stone in it.

from the observers themselves :
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and as a friend of mine put - time travel can be thought about.

My son tells me from US that his erstwhile classmate is now with the LIGO project as also many other Indians. They have translated the LIGO press release into many Indian languages like Tamil, Telugu, Oriya, Marathi and more.

Interesting! I'm pretty sure that the whole 'space-time' fabric thing and the gravitational waves will never pierce through my brain. ☕

This is the IUCAA lecture live stream video on GW shared by director Somak Raychaudhury

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An update:
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