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Spacetime curvature is the relativistic way of describing gravity.

The best way of visualising this (without the benefit of mind-altering substances) is to think a few dimensions down the ladder.

If spacetime were 2-D, then spacetime could be visualised as a elastic sheet.
If you place heavy objects on the sheet, it will bend under the weight, creating a dip.

Passing objects will tend to move closer to the centre of this gravity well, which we experience as gravitational attraction.
The more massive the object, the deeper the gravity well.
Planets orbit stars inside their gravitational fields, like stunt riders on a sloping Wall of Death.
See those things that look like tubes leading straight down from the sheet?
Those are black holes, with their little singularity sulking in the centre.

Happy? Ok, now make the sheet 3-D.
(Most people can't do this).
Now make it 4-D.
(People who can do this don't tend to be allowed near sharp objects.)

Of course, a curved 4-dimensional hypersurface needs a 5th dimension to curve in.
Luckily, math geeks like Riemann and Gauss found you can calculate 4-D spacetime curvature
without needing to know anything about the 5th dimension itself
(which is handy, as we have no way of measuring that)

This works in the same way that Flatland inhabitants could calculate 3-d curvature without being aware of a third dimension.

Hmmm. Ok, visualisation time again:

A. Square and T.Riangle live in a curved 2-D universe
(which to hyperdimensional superbeings like us looks like a sphere).

• They both stand facing North on the equator: one at 0 degrees longitude, one at 90 degrees longitude.
• They both head due North
• Time passses.
• They bang heads together at the North Pole.
Now, (being 2-D), this seems quite odd.
As far as they are concerned, they both set off on a flat surface parallel to each other.
Euclid (who was a Flatland kind of guy) says that parallel lines never intersect.
So our Flatland pair think there must be some wierd force attracting them off course toward s each other.

Once one of them has the trippy idea of curvature in a 3rd dimension,
they can not only explain this without the benefit of gravity,
but they can also figure out exactly how curved the surface is.
The bigger the curve, the sooner their paths will cross.

And it's the same for us.
Just don't try to visualise it when you're sober, and you'll be fine.

You work out the spacetime curvature using what is called the Curvature Tensor, the mathematical tool we use in relativity to explain the geometry of the surrounding spacetime of an object with a dense mass. Usually, the curvature of celestial objects are quite weak, unless we are talking about denser objects like Pulsars or even Black Holes.

Gravity was originally understood (and still is to this day) as a property of mass. Objects made from mass (and also massless radiation as we shall soon see), have the ability to attract other objects with mass. However, Einstein showed that mass was not special when considering gravity, that massless radiation as well could exert a gravitational force (albeit weak) on other objects and he fully described this using what is called the Energy-Momentum tensor; this specific tensor can be written as a matrix and was initiated into his equations which helped describe what is known as the spacetime curvature.

You will need to understand some math for you to fully understand everything written here today, but I will give a briefing of his theory - it will help knowing the basics of Einstein Notation.

The Gravitational field is denoted by Γ ''gamma'' and is called ''the connection'' and is often simply named the Christoffel Symbol after the mathematician who discovered it. Curvature can be fully described by the Riemann Tensor, aka. the Curvature Tensor. A tensor is a mathematical object which can be quite difficult to explain, but the usual explanation is that a tensor is a multidimensional object. The curvature tensor is given as

(For this work, all superscripts are denoted with A^(b) and subscripts as A_(b))

R^(ρ)_(σvμ) = ∂Γ^(ρ)_(μσ) - ∂^(ρ)_(vσ) + Γ^(ρ)_(μλ)Γ^(λ)_(vσ) + Γ^(ρ)_(vσ)Γ^(λ)_(μσ)

You can write this more compactly by rewriting it using the derivatives of the connection.

∂Γ_(μ)/∂x^(v) - ∂Γ_(v)/∂x^(μ) + Γ_(v)Γ_(μ) - Γ_(μ)Γ_(v)

It is the right hand side of Einstein's field equation which describes the matter in the universe

▼_(μ)R_(μv) = ½▼_(μ)g^(μv)R

where g is the metric and ▼ is the covariant derivative which originally came from the mathematical work on fibre bundles. Even if the matter content is zero, this does not imply that there needs to be zero curvature. As explained, we may have various types of energy which can cause the curvature of spacetime. Space-time, to quickly explain what this concept is all about, is the idea that space and time are unified into a single object. Minkowski, who was Einstein's teacher, created the four-dimensional manifold known as spacetime and ever since it's creation it is often simply accepted that space and time are unified in such a way. When someone asked Paul A. M. Dirac (a very notable physicist known for creating the Dirac Equation which successfully describes Fermion particles) whether he believed that the unification of spacetime was fundamental, he said he would chose not to believe this. Today, it stands as one of the cornerstone idea's which have helped shape physics and our understanding of the vacuum.

Anyway, as has been explained, one can have various kinds of energy - one such kind of energy that was predicted in Einstein's theory was the existence of gravitational waves. This was the gravitational distortions that matter would radiate - however, it turns out that not a single gravitational wave has ever been detected, and we have been searching for a good while now, however there may have been some evidence recently (1). It is possible they don't exist in nature. As quantum physics developed it's field theories over the years, field theory was applied also to Einstein's field equations and a new particle was born, the graviton. This particle, which moved at the speed of light and is a Boson, was predicted to exist as it could help explain how gravity is mediated from one location to another - however, not a single graviton has been found in nature either. The truth of matter is that our understanding of gravity has not been drastically enhanced since the development of General Relativity.

It is possible that gravity is a psuedoforce - a ficticious force similar to the Coriolis force. A psuedoforce is not a real force in the sense of electromagnetism or the weak or strong forces. It would not require a physical mediator particle - it is a special kind of force experienced by frames of reference.

Today, gravity is still understood in terms of Einstein's beautiful theory, that is the source of gravity is the geometry of the vacuum, intrinsically related to acceleration, matter and energy.

(1) - http://www.bbc.co.uk/news/science-environment-19408363

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