General relativity is just a fancy 20th-century name for gravity – the force that pulls

stuff to the ground and keeps the planets in orbit around the sun .

General relativity is the idea that gravity happens because space is curved – like how

when you walk along the surface of a ball, you end up curving downwards.

Actually general relativity is the idea that space and time aren't separate – time

is a physical dimension, and together they form a single geometry called spacetime – and

gravity is caused by curvature in spacetime.

Except, curvature in spacetime on its own doesn't explain gravity – I mean, just

because you're on a curved ball that doesn't mean you follow a specific path on the ball.

General relativity is the combination of the ideas that spacetime is curved AND that stuff

in spacetime obeys laws of motion : an object in motion stays in motion along a “straight”

path in curved spacetime . Like following a straight line along the surface of a ball.

But spacetime can't be curved any which way.

Just like how a ball looks flat when you're close enough, curved spacetime is locally

flat (rather than crumpled, or something).

Of course, flat space makes sense, but what does it mean for spacetime to be flat?

That it obeys the rules of special relativity!

Finite speed of light, time dilation and length contraction, relative addition of velocities,

and all that.

Basically, if general relativity is like a globe, special relativity is being on the

surface of the globe (and mathematicians call the globe a pseudo-Riemannian manifold with

Lorentzian signature).

Except we still haven't said anything about what determines the curviness, or shape, of

spacetime in the first place!

Or why objects tend to follow straight paths in that spacetime.

General Relativity is actually the idea that all the stuff in spacetime – matter, radiation,

pressure, energy, momentum, particles, and so on – all that, together with spacetime

itself, obeys a set of equations called the Einstein Field Equations.

These equations look simple if you write them in a clever way, but they're actually a

very complicated set of ten nonlinear differential equations [2nd order] that you have to solve

in order to make predictions about how spacetime will curve and how the stuff in it will move,

depending on how spacetime is curving and how the stuff is moving .

The solutions to the Einstein Field Equations of General Relativity describe gravity around

solitary objects like black holes or the sun or the earth, and they facilitate very accurate

predictions of orbits around these objects . But these equations aren't limited to

describing just energy and matter and spacetime around the earth or sun – they can be used

on the universe as a whole to describe and understand the past and present and future

of the cosmos.

So.

General relativity is the idea that the universe can be described by a pseudo-Riemannian manifold

representing spacetime and an energy-momentum tensor representing all matter and energy,

which together obey the Einstein Field Equations.

For our 3+1 dimensional universe, the predictions generated by this idea have been experimentally

verified by many many incredibly precise observations, ranging from the precession of the orbit of

Mercury, to the slight drift of the moon's orbit away from the earth, to the gravitational

lensing and redshift of starlight, to time dilation of atomic clocks and precession of

gyroscopes orbiting the earth, to observations of the polarization of the cosmic microwave

background radiation, to gravitational wave detections of black hole mergers, to direct

imaging of the black hole at the center of the Milky Way.

Wait but how does general relativity explain the everyday “force” of gravity we experience

on earth?

Well, you know how when a vehicle turns but your body's inertia makes you want to go

straight and it feels like you're being pulled sideways because you're being accelerated

away from your straight-line path?

In general relativity an object's straight-line inertial path is actually to fall towards

the center of the earth, and since the surface the earth accelerates us away from that straight-line

path, we feel that acceleration as a weight or force that we call gravity [equivalence

principle].

If you're in free fall (or in orbit), then you are following a “straight path through

curved spacetime”, that is, you're not accelerating in spacetime, so you feel like

you're floating or experiencing “0 g”.

And that explains gravity!

Oops.

General relativity doesn't explain quantum mechanical phenomena, and has problems jiving

with the theory of quantum mechanics in certain extreme situations.

Physicists have been working for over 90 years to reconcile general relativity and quantum

mechanics in those situations, and while we've learned a lot, we still haven't come close

to solving everything!

It's hard.

General Relativity works so well in most cases where we can test it, and quantum mechanics

works so well in most cases where we can test it, and they're both so mathematically sophisticated

and constrained and distinct from each other, that imagining a different mathematical model

that encompasses both while being just as accurate where they're already accurate

and better where they're in conflict is… very high level stuff.

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