Being a technological civilization means beaming "Sorry for the delayed response – finite speed of light!” back and forth with astronomers circling a distant, lonely star until one of you manages to extinct yourself.

Prime numbers were chosen for the dimensions of the array, in the hope that they would be conspicuous to potential recipients.

The message should arrive in about 22,000 years, give or take a few hundred. So in around 44,000 years, maybe our futuristic detection network will pick up a delicate radio whisper in response.

Maybe we'll still be around to decode it. Maybe not.

The "Arecibo Message" was beamed towards globular cluster M13 #OTD in 1974, during the dedication of an upgrade to the radio telescope. It was the first message sent with the intention of alerting extraterrestrials to life on earth.

The message was 1679 bits, arrayed into 73 rows and 23 columns. It contained depictions of the numbers 1-10, the solar system, the human form, and info about the elements and biochemical structures associated with life on Earth.

(I am writing up notes and exercises for a group of students doing an independent / directed study course on general relativity. It would be easy to lose myself in this, and spend the whole day / week / month on it.)

If you view the entire shaded region (orange and green) as its own spacetime, the interpretation is a wormhole connecting two distinct regions that are both "outside the black hole."

However, this sort of wormhole isn't traversable. An observer has to move along a path that, at each instant, remains inside their light cone from the instant before.

Light rays make 45 degree angles on this diagram, so an observer starting on the orange side can't cross the 45 degree line into the green region.

As long as I'm posting diagrams, here is the Kruskal diagram for the Schwarzschild spacetime. Hyperbolas (blue) are constant r, and straight lines (red) are constant t. The 45 degree lines (purple) crossing the origin are the horizon, and the bold hyperbola (black) is the singularity.

Schwarzschild's coordinates cover the gray patch, Eddington-Finkelstein coordinates cover the orange patch, and Kruskal-Szekeres coordinates the entire (orange + green) shaded region.

Albert Einstein used my signature move #OTD in 1915. He pretended to be sick so he could skip a seminar by his colleague David Hilbert, then stayed home to work on his own theory.

(From "David Hilbert and the Axiomatization of Physics" by L. Corry)

In the region r < 2M, all the light cones have tilted over so that they lead inexorably inward.

Any observer in this region, geodesic or not, must move towards the singularity at r = 0 (the thick black line). There simply are no escape routes to larger (and safer) r.

What a remarkable thing! It's not like you could escape if you just had a more powerful rocket. The thing that traps one inside a black hole is *causality*.

When r > 2M, the observer has the option of heading in a direction of larger r, away from the black hole. This is accessible within their light cone at that instant. They just fire their thrusters, or whatever, to get off that infalling geodesic and fly away.

But once r < 2M, they cross into a region where this is no longer an option.

Outside the black hole (r > 2M) there are two sorts of radial null geodesics: light rays can follow a path that moves outward (towards larger r) or one that moves inward (towards smaller r).

Inside (r < 2M) the two sorts of null geodesics both head inwards, towards smaller r.

Notice that at each point along the infalling observer's worldline, they find themself inside a "light cone" defined by the two sort of radial null geodesics emitted at their previous location, the instant before.

Black holes!

The plot below shows two observers in the Schwarzschild spacetime. One falls inward along a timelike radial geodesic (purple), another (green) remains at a fixed value of the radial coordinate r.

The infalling observer emits radial signals (red) at equal proper time intervals. These signals move outward along null geodesics, and the stationary observer receives them with decreasing frequency.

Well well well, what do we have here. If it isn’t Murderbot, and what appears to be a second murder-oriented bot.

The result has since been refined by measuring the effect with distant space probes.

In 2003, Bertotti, Iess, and Tortoran used a signal sent to and from the Cassini probe to confirm the predictions of general relativity with an accuracy of 20 parts per million.

https://www.nature.com/articles/nature01997

Shapiro et al tested the idea in 1966-67, first with Venus and then with Mercury, at MIT's Lincoln Laboratory Haystack radar site.

The results agreed with the predictions of general relativity to well within the expected uncertainties. #TeamRadar

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.20.1265

Shapiro estimated that signals bouncing off Venus or Mercury at superior conjunction, when they are on the opposite side of the sun from Earth, would experience a delay of about 200 microseconds due to the Sun's gravitational field.

This was a measurable effect at the time!

It's a simple idea: The round-trip time of a radio signal bounced off a distant object increases a little if it passes through the gravitational field of a massive object along the way.

Image: Simon Tyran, https://en.wikipedia.org/wiki/File:Shapiro_delay.gif

When Einstein proposed general relativity he laid out three tests: the precession of Mercury's perihelion, deflection of light by the sun, and gravitational redshift.

Irwin Shapiro proposed a fourth test #OTD in 1964: the gravitational time delay of light.

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.13.789

@roboconnor Just got back from watching it with my 11yo. We loved it and had a blast.

Good morning.

Sometimes these really hit home.