## What is an inverse meter?

The relationship between Wavelength and Frequency is an inversion.

It reminds me of the relationship between Radius and Curvature.

$wavelength = \dfrac{1}{frequency}$

$radius = \dfrac{1}{curvature}$

When one gets bigger, the other gets smaller, and vice versa. In fact a circle with zero curvature is called a “line”, and some people discuss it’s radius as being infinite.A circle with zero radius is sometimes called a “point” and it’s curvature is considered by some to be infinite. Similarly with Spacetime, it’s curvature is considered by some people to be infinite at the center of a Black Hole.

Wavelength and radius are both technically measures of the separation of points in space.

When talking about light in spacetime,

$wavelength = \dfrac{speed \, of \, light}{frequency}$

For example the light coming out of the computer screen as you may be reading this, is roughly $10^-6$ meters wavelength, and $10^{14}$ cycles per second,  ( ScienceForums, jhuapl.edu). Or, a sound of note on a piano is roughly 330 cycles per second, with a wavelength of, well, say sound is roughly 700mph, 5/3 km per mile, 1200 kmh, 1200km/3600seconds, 1km/3s, 1000m/330*3, roughly 1 meters.  (for exact, see liutamottola.com). For an ocean wave, the frequency might be 1 cycle per 30 seconds, with a wavelength of a few hundred meters.

How would this relate to a radius and curvature?

To start with, radius is in unit of distance, like wavelength. But Curvature- what are the units of curvature?

We could start with something basic, “1 over a meter” or “inverse meter”. But what does that mean?

If I have a circle of radius 1, the curvature of that circle is also 1. 1/1 = 1. If my circle is radius 2, then the curvature is 1/2.

But then, curvature is depending on the units I am using. And so does it actually tell me anything?

If I say that I have a circle of radius 1 meter, I can picture that in my head. Perhaps it is a very large hoola hoop, or maybe it is a jump rope placed on the ground end to end in a circle.

If I say that I have a circle with curvature  of one inverse meter, what does it mean? What if I say I have a circle of 2 inverse meters? Does that tell me anything?

Sure, at first glance, I can convert curvature to radius in my head, and then picture a circle of about that size. But is there something else I can think of? After all, if I picture a radius, and  circle based on a radius, then I am not really picturing “inverse meters” am I?

This is the difference with Wavelength vs Frequency. I can picture both – I can imagine hearing a song, and knowing whether it is higher in pitch or lower, in other words whether it is high frequency or low frequency. I can picture an instrument like a Tuba or Trombone vs a Piccolo Trumpet or tin whistle, and see that longer wavelength gives lower frequency. I could even get a guitar, and place my fingers on the frets, and notice that the longer the vibrating part of the string, the lower the sound. And the wavelength is right there – it’s directly related to how long my string is on my guitar.

I can see a spinning wheel and know whether it is spinning fast, at high frequency, or slow. I might even make a guess as to the speed, such as revolutions per minute.

But what is an Inverse Meter?

Hold on a second. 15,00,00,000 as they might write in India. 15 crore? At any rate

That means the curvature would be 150 million inverse kilometers.

If I had a running track to go around and it was 110 meters across, I would have a running track with curvature of 110 inverse meters.

If I had a tea leaf swimming around the inside of my tea cup, it would have a curvature of 0.1 inverse meters.

So perhaps I can visualize this thing called an inverse meter… maybe I can think of it as the ‘straightness’ of the path I follow if I were to travel around a circle?

I kind of like that. It reminds me a little bit of my attempt to read the book “Gravitation” by Wheeler, Thorne, and Misner, where they explain Einstein’s view of General Relativity.

Then again, that “straightness” is relative to my size! For an ant, the inside of a teacup might seem as to me a soccer stadium. And to a giant space being, the orbit of the Earth might seem as though the inside of a teacup.

But distance itself, is also relative to one’s size is it not? To me, the distance to the moon seems a far stretch, but from the perspective of a giant being, it seems short. To me the distance to my bicycle seems short, but to a microbe it might be a lifetime’s travel.

And maybe this idea has been with us a long time. I might say that I travel a long and winding road, to indicate it took a great effort to get somewhere. This shows that there are two aspects of tribulation, the distances of the paths, and the curvatures. A straight path, the Straight and Narrow, takes one to a destination more quickly. The shortest distance between two points is a straight line. I.e. a circle of zero curvature.

But we never developed a nice unit for curvature, only one for distance. What a shame!

Or did we?

We have the steepness of a slope. We have the sharpness of a bend. We have the Grade of a hill. Are these measuring the curvature of the Earth in some sense, perhaps?

In the relativity view of the world, Gravity itself is simply the curvature of spacetime. What units, I wonder, do the physicists speak of when they say that? How do they measure it? Do they have a funky ruler that measures spacetime curvature?

Gravity by some accounts is measured in G-forces. When you drive over a hill too fast, you have negative gs. When you drop on a rollercoaster, you have negative gs. When you come to the trough of the coaster, and are pushed into your seat, you have positive gs above 1. Standing in a field, there is only 1 g.

Can you say that g is a measure of curvature then?

After all, one does not typically speak of the “radius of the track bend” on a roller coaster, instead one discusses how many g forces it exerts on you, as a passenger. But… those gs are not just dependent on the bend of the track, but on the speed of the coaster as it enters, which, in turn, is effected by the number of passengers in the ride cars, their weight, windspeed, track surface conditions, etc.

But what of an accelerating automobile on a straight track? It pushes you horizontally back into your seat, and that too is measured in gs. But where is the gravity in this case? You would experience this acceleration feeling even if you were in space, with some booster rocket going off, pushing you back into your seat, with no planet to be found anywhere nearby.

What curvature is that?