Spacetime is Curved

“General relativity can be summed up in two statements:

1. Spacetime is a curved pseudo-Riemannian manifold with a metric of signature .
2. The relationship between matter and the curvature of spacetime is contained in the equation
   .”

—Sean Carroll

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There you have it, ladies and gentlemen! The universe summed up in two sentences.

Regarding the first statement: a metric of signature just means that time goes the opposite direction as space. (Think light cones.)

No matter how fast you go — or if you stand still — time is flowing through you at the speed of light.

Another way to think about going somewhere, say Bermuda, is that you will meet up with Bermuda’s future. To hurry to Bermuda is to meet an earlier future — but no matter how fast you go, the Bermuda that is now will be gone by the time you get there.

And so we beat on, boats against the tide, borne back ceaselessly into the past.

The universe is a song, singing itself.

Maybe Calculus Isn’t Bullsh*t

I may have to take back what I wrote earlier. I need to find out more about the Radon-Nikodym concept of a derivative.

Weather

Predict the weather for the coming week by looking at the whole continent. If the jetstream is above you, you’ll get warm Mexican air. If the jetstream is below you, you’ll get cool Canadian air.

(Works if you live in the midwest. Of the US.)

Current National

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Current Visible Satellite

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Current Infrared Satellite

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Current Temperatures

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Current Northeast Temperatures

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Current Wind Chill

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Current Heat Index

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Current Dewpoints

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Monday Jet Stream

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Tuesday Jet Stream

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Wednesday Jet Stream

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Thursday Jet Stream

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Friday Jet Stream

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Saturday Jet Stream

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Sunday Jet Stream

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Today’s Lows

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Today’s Highs

My dad made this.

The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that mathematicians have made a covenant with the devil to darken the spirit and confine man in the bonds of Hell.

St Augustine

This new Andrew Bird jam makes me excited about Andrew Bird again. (It’s not his fault that I and people around me over-played Dark Matter to its grave.)

Hot Math by Andrew Bird

In a space of high dimensionality, most of the volume of a cube is concentrated in the large number of corners, which … become very long ‘spikes’!

(Reason being: The ratio of the distance from the centre of a D-dimensional cube to one of the corners, divided by the perpendicular distance to one of the sides, is √D, which therefore goes to ∞ as D →∞. )

Neural Networks for Pattern Recognition by Chris Bishop

Problems 1.18, 1.19, 1.20

Machine Learning and Pattern Recognition by Chris Bishop

The best determinant of someone’s level of happiness is how often they eat meals with friends and family.

Would You Be Happier If You Were Richer?

6-dimensional Sphere

Of course a 3-dimensional thing is bigger than a 2-dimensional thing. Just like a beach ball is bigger than a circle cut from paper.  And it’s equally obvious that a 4-dimensional thing would be bigger than a 3-dimensional thing.

Or … is anything obvious? Following is a lesson in multi-dimensional reasoning.

A 2-dimensional “ball” is a circle. Two-dimensional points with two coordinates comprise the 2-ball. Any pair of rectangular coordinates on the circle satisfy:*

.

For example the pair (√½, √½) is on the circle. And the pair (√¼, √¾) is also on it. But (½, ½) isn’t. 

A 3-ball is made up of the 3-points whose three rect-coords satisfy

.

That’s just a normal sphere from real life.

And a 4-D ball would consist of any array of four values whose Pythagorean sum** is 1:

A 5-ball is likewise the set of all 5-points with Pythagorean norm 1. It’s bigger than a 4-ball, of course.

A centered unit 6-sphere is the set of all 6-points with norm 1. It’s smaller than a unit 5-sphere.

• You: What?
• Me: Don’t ask what, you heard me.

A 6-dimensional ball is smaller than a 5-dimensional ball.

The 7-ball is smaller still, and higher dimensions keep getting smaller in volume.

True story.  I am not making this up.

If you don’t believe me, just do the sextuple integral. The volume of a 6-sphere is 5.6771 and the volume of a 7-sphere is 4.7247. A 13-ball is less than 1 unit volume.

* I’m talking about a “unit circle” with radius 1, but that could be radius 1 mile or radius 1 nanometer. Or, like, whatever.

** Pythagorean sum? I’m being sly. Hinting at future posts about

• L_p norms
• √2
• Pythagoras in higher dimensions
• statistical variance as a Euclidean phenomenon.
• the logic of multiple dimensions