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hpr3126 :: Metrics part II

The metric of a 2D curved surface

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Hosted by Andrew Conway on 2020-07-27 is flagged as Clean and is released under a CC-BY-SA license.
Tags: maths,relativity,physics,earth.
Listen in ogg, spx, or mp3 format. | Comments (3)

In this show I continue from where I left off in my last show (3101) and talk about the geometry of curved 2D surfaces such as that of a sphere.

Using the Earth's surface as an example we can use familiar the co-ordinates of longitude and latitude, illustrated on this diagram:

Sphere with latitude and longitude

Source: Public domain, Wikimedia commons

On the left we see circles of constant latitude. The largest of these circles is at latitude 0° and is called the equator. Its circumference is equal to that of the Earth and so it is an example of a great circle.

On the right we see lines of constant longitude. These run from pole to pole and are perpendicular to the equator. Each of these lie on a great circle (in fact they are half a great circle each).

Great circles on the surface of a sphere are analogous to straight lines on a flat 2D surface. They offer a way to connect any two points with the shortest distance. Lines in 2D or great circles on a sphere are examples of what is called a geodesic. In physics, particles that are not subject to any forces will follow geodesics.

In Einstein's General Theory of Relativity, the presence of mass or energy will alter the shape of spacetime and that will determine the metric. From the metric you can derive the geodesics and from that you can predict the motion of objects with no forces acting on them. In this way you can do away with the approximation that is Newton's gravitational force and replace it by a description that only involves the curvature of spacetime. I only touch on this in this show but will likely return to it in future shows.

Here are the equations discussed in this show and the previous one:

Sphere with latitude and longitude

Comments

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Comment #1 posted on 2020-07-29T13:29:26Z by Dave Morriss

Etymology of 'geodesic'

Hi Andrew,

Regarding 'geodesic' I found some answers here:
https://www.etymonline.com/word/geodesic

So, 'geodesic' is from 'geodesy' which means "surveying".
Then 'geodesy' is built from 'ge' (pertaining to the Earth) and 'daiein' meaning "to divide".

This is just my limited summary, see https://www.etymonline.com/word/geodesy for the fuller definition.

Enjoying the series; more please!

Dave

Comment #2 posted on 2020-07-31T22:41:39Z by sesamemucho

Special thanks

I've enjoyed listening to your last two shows. You presented a very clear introduction to the concept. It's amazing how such an apparently small topic can unfold into something so complex and interesting.

On a tangent, I'm doing exactly this calculation for a project at work, and this episode showed me a simpler and more easily explained way to do it.

Comment #3 posted on 2020-11-25T16:43:29Z by mcnalu

Thanks for the comments

Dave - thank you for looking that up. Dividing surprises me actually as I was thinking it was more to do with movement but lines divide a 2D space and great circles divide a sphere so it makes sense.

sesamemucho - glad it was helpful. "On a tangent..." is a very apt choice of words!

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