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Newton conceived space as being:

Absolute space, in its own nature, without regard to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies: and which is vulgarly taken for immovable space

and motion as

Absolute motion is the translation of a body from one absolute place into another: and relative motion, the translation from one relative place into another

But space cannot be absolute as there is no point of origin, every point being similar to every other.

Hence space itself shouldn't be conceived at being at rest.

This is in fact, in a different guise, an argument offered by Leibniz to argue that space in itself is relative.

Does this work as a motivation for gauge freedom of space?

The argument appears to give space gauge freedom locally, whereas it is global gauge freedom that gives particles laws - laws of conservation.

What would it mean, physically for space to have gauge freedom locally?

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    Space is not a substance; therefore, how can it move or be at rest? – Geremia Apr 16 '14 at 1:56
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    if space isn't a substance then how can it have curvature (for gravity)? – Mozibur Ullah Apr 16 '14 at 1:59
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    That's an abstraction used to describe the effects of gravity. – Geremia Apr 16 '14 at 2:40
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    That space warps is real enough - you have what is called gravitational lensing which is viewable by telescopes. If our eyes were good enough we'd see it too. On a hot summers day we see the scene before us shimmer. Its the same effect. More philosophically if space isn't a substance then how does gravity travel from the Sun to the Earth - is that an abstraction too? Does the force of gavity somehow teleport itself from the sun to here magically? This is one reason why there were mechanical theories of the transmission of the force of gravity in the 18-19th C, if not earlier. – Mozibur Ullah Apr 16 '14 at 3:28
  • It's space-time that warps. Sure, gravitational lensing is observable, but as space-time curvature and gravitational force are the same thing you can just view it as the effect of a force not unlike those found in any other kind of lens. Does space (without the -time) warp in general relativity - hard to say - it is not uniquely defined. – Lucas Apr 16 '14 at 11:02
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This language is undoubtedly influenced by the ideas of Galileo, what, in modern times, is called Galilean invariance. Using modern language: this is the global symmetry under which the laws of motion (i.e. Newton's laws) are invariant. This symmetry persisted as a foundation of mechanics up until the time of the discovery of Relativitity, when Lorentz invariance was recognized as the relevant symmetry property. Although General Relativity does cover the cases where there are locally defined reference systems, this is not (to my knowlege) referred to as a gauge symmetry -- that term is reserved for the complete invariance of the dynamics, in particular the invariance of the solution(s), to changes in some of the degrees of freedom in the system description.

Section IV of the treatise further elaborates the idea of relative space. Its unclear to me, who is not a scientific historian, to what extent he is wedded to the idea of the idea of "fixed space", although he does not explicitly reject it. Newtons elaborations on the movement of the ship relative to the Earth, and things in the ship that move relative to it, is a clear reference to Galilean invariance. I find it interesting that he hedges on whether the Earth itself moves in an absolute sense -- either he believed that this was a possibility, or he recognized that, for the purposes of any one discussion, you can pick a particular reference frame, and measure all motions relative to it; making that selected frame the arbiter of what constitutes absolute motion.

Note that all symmetries lead to a conservation law (Noether's theorem) it's just that in the case of local symmetries, that convservation law is a conserved current.

In one sense your final question doesn't have an answer: there simply are not gauge degrees of freedom for spacetime itself in GR. The other way to look it is that GR provides the description for locally varying frames of reference already, but it is not a gauge theory.

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Newtons work is hugely important but it has been improved upon and largely superceded by that of Einstein and Hawking once you leave the field of classical mechanics. In Einsteins model, the three Euclydian dimensions of Space and the fourth dimension, Time, combine into a single four dimensional continuum known as Minkowski Space. This makes it possible to desribe supergalactic and subatomic processes in a more complete and uniform way.

The mutable nature of space-time is observable in the way that it is distorted by the gravity of massive objects such as stars and planets. This effect is known as gravitational lensing and is well proven with observational evidence.

From NASAs Image The Universe (http://imagine.gsfc.nasa.gov):

There has been experimental evidence for the curvature of spacetime by a massive object since the early part of this century (1922), when observers set out to test the predictions of general relativity. During a solar eclipse, they realized, the light from stars in the same general area of the sky as the Sun are visible during the day. If light from these stars is affected by the curvature of spacetime due to the Sun's mass, then this would be measurable as a deflection (or a change in location) of the star's position on the sky. The stars closer to the position of the Sun in the sky would suffer a larger deflection; in general the deflection would be proportion to the stars distance from the Sun's location on the sky. This effect was observed for 15 stars during the solar eclipse of 1922 in Western Australia, and was interpreted as observational verification of the predictions of general relativity. General relativity predicts that spherical masses deform spacetime in much the same way a lead ball would deform the surface of a rubber sheet. It is this deformation that causes the planets to orbit the Sun, and the Moon to orbit the Earth. In fact, all orbital motion is the result of bodies being affected by the curvature of the spacetime in which they move. Since that time, astronomers have observed other instances of the curvature of spacetime near massive objects. One example is the deflection of radio waves from quasars which are occulted by the Sun every year (such as 3C 279). Another is the growing collection of gravitational lenses. A gravitational lens occurs when the light from a very distant object (often a quasar) is bent by a closer massive object (such as a galaxy) into multiple images. Some very impressive images of gravitational lenses have been taken.

From this we can say that with a more current understanding of space, it is not 'at rest' in the absolute sense that Newton meant as it can be moved and influenced by the presence and passage of objects passing through it. This has been postulated as a possible method of FTL travel: The folding of space into a 'wave' and riding said wave like a surfer (This is how the warp engines of Star Trek work) or folding space over on itself to make the distance between origin and destination zero and staying at the destination when space snaps back into place (like the engines from Event Horizon). Both are theoretically plausible methods of FTL travel (but impossible with current technology).

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