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It seems to me that there is considerable relationship between the three concepts: frame of reference, observer, and gauge. How do they overlap?

My current understanding is that an observer with a measuring apparatus (or in particular, a Euclidean space) is a frame of reference. And that a gauge is a "continuous" set of local measuring apparatus associated with a single observer. Would that be far from the truth?

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A frame of reference means a co-ordinate system and an observer is someone using that co-ordinate system. For example I could define cartesian co-ordinates (t, x, y, z) and define myself to be at the point (0, 0, 0, 0) and since I'm always at the origin that means I'm at rest. So when I measure your position and speed I'm the observer and I get some value in my frame i.e. my co-ordinate system.

The distinction between frame of reference and co-ordinate system may seem rather minor, but when you get on to general relativity you'll realise the importance of considering co-ordinate systems. For example when calculating the properties of black holes we commonly use three co-ordinate systems, Schwarzschild co-ordinates, shell co-ordinates and freely falling co-ordinates. None of these would normally be described as a "frame".

In physics a gauge has a rather special meaning. It's a reference point used for some set of co-ordinates. For example suppose I want to measure the height of Mount Everest, the question is what should the height be relative to? Generally we'd measure the height above sea level, but you might just as well measure the height relative to the centre of the Earth. The reference point we choose defines our gauge, and in choosing the reference point we are fixing our gauge.

Gauge invariance is a very important part of modern physics. The height of mountains is a rather trivial example, but it's hard to find an area of physics that gauge invariance doesn't affect. See http://en.wikipedia.org/wiki/Gauge_fixing for more info on gauges, but be warned that article is a bit technical.

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A gauge just sets the zero point of your "Measuring device" –  Argus Jun 8 '12 at 22:13
    
Thanks. And simplistically speaking, gauge invariance would be when the results are not dependent on the manner of gauge fixing, correct? –  Muhammad Alkarouri Jun 9 '12 at 16:36

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