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Q & A: cosmic questions

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Most recent answer: 10/22/2007
Q:
1) Was Einstien the first to say "laws of physics are the same everywhere in uinverse.", else who? 2) Is that a priciple -- that we accept until proved otherwise -- or does it have a proof -- what is the proof? 3) A curved space (such as the surface of a sphere) has different values for pi, depending on the radius -- distance from the pole along a geodesic. The maximum of pi occurs on a flat plane -- 3.14..... One can prove this value (3.14) mathematically -- using the limit process and some trigonometry. Is there a physical reason for this value, i.e., what property of space leads to 3.14 value being the maximum in our universe?
- Mehran
Miami
A:
Mehran- Welcome back.

1. The idea that the laws are the same everywhere goes back to before Newton, but I don't know who was the first to make it explicit. Even the idea that the laws should look the same to observers in constant motion with respect to each other is old. It was first clearly stated by Galileo, and put in more precise form by Descartes.

2. It's a principle that fits with many cosmological observations. Generally speaking, we don't do proofs of basic principles. We just see if they work.

It's also a practical feature of what physicists do when seeking to understand what the laws of physics are. If a "law" is found to describe nature in some places but not others, it loses its status as a "law of nature" and at best becomes a "local approximation". Instead, we'd try to find out what it is that makes stuff behave differently in one place than another, and formulate laws that describe the behavior everywhere under as common and general a description as possible. One very famous example of this is Newton's realization that the same law of gravity applied to objects falling on the Earth as makes the planets go around the Sun and the Moon around the Earth.

3. Really on any space whose curvature is finite, one gets the same limiting circumference/diameter ratio, our old pi, for small circles.

For other circles, the ratio can be smaller (say on a sphere) or bigger (say on a saddle).

And beware of the 'proof'. Are you sure that your assumptions (trigonometry) don't already contain the thing you want to prove?

I'm not sure how to answer the question about "why 3.14159...etc?" The reason that you get the same vale for all sufficiently small circles is that if the curvature is finite then on a small enough scale everything looks like flat space. Then similar figures exist on different scales. The ratio has to then come out to be some particular number, and any rigorous argument giving how to obtain that number (going back to Archimedes at least) is an equally good way of explaining it.

We're not too keen on redefining pi based on the curvature of space and how big your circle is. It's better to leave pi as the number it is, and just say that the relationship of the circumference of a circle to its diameter involves other numbers than pi in curved space. (pi has lots of other uses than just talking about circles and spheres, although I suppose they are all related in some way).

Mike W. and Tom

(published on 10/22/2007)

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