Silly question about Galilei Group

Question

I have an silly doubt about Galilei Group.

From Wikipedia:

"The Galilean symmetries can be uniquely written as the composition of a rotation, a translation and a uniform motion of spacetime. Let x represent a point in three-dimensional space, and t a point in one-dimensional time. A general point in spacetime is given by an ordered pair (x, t).

A uniform motion, with velocity v, is given by ${\displaystyle ({\mathbf {x}},t)\mapsto ({\mathbf {x}}+t{\mathbf {v}},t),}$ where $v ∈ ℝ^3$.

A translation is given by ${\displaystyle ({\mathbf {x}},t)\mapsto ({\mathbf {x}}+{\mathbf {a}},t+s),}$ where $a ∈ ℝ^3$ and $s ∈ ℝ$.

A rotation is given by ${\displaystyle ({\mathbf {x}},t)\mapsto (G{\mathbf {x}},t),}$ where $G : ℝ^3 → ℝ^3$ is an orthogonal transformation."

What suppose to mean, physically, the $t+s$ (I mean: $t'=t$?) and $\textbf{a}$ (on $\textbf{x+a}$) on translation? And why we are considering rotations (I mean, these kind of motion are non inertial)?

Answer 1

The translation just adds a constant vector a to any vector in 3d and a constant time shift $s$ to the time $t$. This amounts to translating the origin in space to -a and the origin of time to $-s$.

The rotations are time-independent and are just fixed rotations taking one set of axes to another, i.e. they are fixed rotations between basis vectors of two inertial frames; basically there is no reason to suppose that the inertial frames have basis vectors that are aligned.

If the rotation was time-dependent, i.e. if the set of basis vectors in the second frame rotated in time w/r to the first frame, then the second frame would not be inertial.

Answer 2

Actually it is a fair question, and even though it is a conceptual and basic question, it is well stated and precisely written. It makes clear what you are confused about. It's two factors, one is thinking of active transformations instead of passive ones (see below)[and the two are equivalent once understood], and the other one is, thinki g of the transformations, at least for G, as a continuously rotating body.

First, the most obvious, the rotation G is not rotational motion (which would have you in a non-inertial reference frame, as you indicated). It is instead a simple rotation of the axis of reference, where your motion is described in the statically defined rotated frame. Ie, you can draw the reference axis with any direction being the x axis, and just rigidly draw the other two so that it is a rotation from my x, y and z axis.

[hopefully not to confuse you, but what I described is a passive rotation, i.e., the reference frames, and what you described was active instead, but also continui got rotate. You could have just rotated the body and anything else in your system, a single rotation and not a continuous one, and still describe it as inertial]

The second question of why and what is s, also shows the same confusion. Think of it as not going to the past or future, but recalibrate get your clocks (or calendars) so that t=0 in the new clock is t=-s in the old one. That's a passive transformation. You can make it active by considering t=s in the new clock as t=0 in the old clock (coordinate system with that clock; or alternatively do t-s).

So the best way to get in-confused is to think of coordinate transformation for a new coordinate frame (=system).

For a continuous rotation you'd need the elements of G to be functions of t, where instead of a fixed rotation by $\theta$ It would have been something like $\omega t$.

Once you understand the passive coordinate transformation it's not too hard conceptually to consider them active, they are equivalent.

Hope this helps, and I can assure you that others have had the same silly confusion.

Answer 3

The transformations you're thinking of as "Galilean Transformations" are, more properly, termed the Galilean boosts; while the others are translations and rotations. A boost is a change from one inertial frame to another that is moving with respect to it - those are the transforms you're not asking about.

The ones you're asking about are the rotations, spatial and temporal translations. In this context "rotation" does not mean setting anything into spinning motion, but means reorienting it. Together with spatial translations, rotations comprise the classical transformations of Euclidean geometry.

Two figures, in Euclidean geometry, are considered congruent exactly when either can be moved onto the other, so as to coincide with it, by repositioning it (spatial translations) and reorienting it (rotations). Depending on which geometry author you read, taking mirror images may also be considered as a congruency, though it really shouldn't. A left-handed trefoil knot and right-handed trefoil knot cannot be moved, either into the other. In 2D it's sorta okay, since you can move a 2D figure into its mirror image by flipping it over in the third dimension; the same for 1D figures. But you can't flip a left hand to become a right hand, or vice versa ... unless you're in the middle of an episode of The Outer Limits and have a fourth spatial dimension available. (That was in one of the episodes.)

The definition of "congruence" by "transform to coincide" is the original approach to the subject that Euclid, himself, took in his ancient monograph on Euclidean geometry.

If you allow change of scale as a transformation, then the corresponding concept lightens up from outright "congruence" to "similarity". Two figures are similar if they can be rescaled so as to become congruent.

The laws of physics are not scale-invariant, but the forces of electromagnetism and gravity are (almost?) scale-invariant. That's why, for instance, if a moon were to orbit Jupiter at such distance that Jupiter looked to be the same size in the sky as the sun does in Earth's sky, then (since Jupiter and the sun have about the same density) the moon's orbital period would be almost exactly one Earth-year. Kepler, by the way, actually was starting to apply his laws to the then newly-discovered moons of Jupiter, but failed to notice this!

Time translation provides a similar kind of "congruence" for the 1-dimensional geometry of time sequences. So, you could say that two sequences are temporally congruent if either could be made to coincide with the other by suitable repositioning in time. For example, the histories of Japan and Britain come pretty close to being temporally congruent. They each have their own version of "we will fight them on the beach, in the hamlets and never surrender", their own versions of languages derived from a mixture of many substrata and superstata, their own version of being saved from conquest by a massive armada by "divine wind".

The fundamental laws of physics are abstracted out from the particulars of any point in space and time and, therefore, are independent of position and time; so that they should be invariant under space and time translations.

They are also independent of direction - which is actually a step away from traditional folklore science that includes such laws as "what goes up must come down" that calls out a specific direction in space. When you restate it, instead, as "what goes away from the Earth falls back [if it didn't go away fast enough] regardless of how the Earth, itself, is oriented" then it becomes independent of direction.

The six degrees of transform making up the Euclidean transformations go with spatial geometry, the one degree of transform making up the time transform goes with temporal geometry, while the boosts - which Galileo invoked as equally fundamental transforms - provide the foundation for the marriage of the two geometries into a single chrono-geometry: a geometry for space-time, as the transform involves a combination of both geometries.

Ironically, that the invocation of boost transforms actually already married space and time into spacetime (before Minkowski's famous 1908 announcement of their marriage) was not recognized before the 20th century as such, so it might be more properly termed an elopement.

Newton, himself, rejected Galileo's supposition. He assumed that there is a fixed frame that is at absolute rest, but that it is unknowable to mere mortals. This is why his first law was stated in two parts: one for systems at rest, and the other for systems in motion. When he said "at rest" and "moving" he meant "at absolute rest" and "in absolute motion".

Newton formulated his laws in such a way as to make it impossible to distinguish - from the laws alone - which frame was "actually at rest" and which ones were in "absolute motion". In other words, he tried to have it both ways: to reject Galileo's relativity principle and then sneak it in anyhow.

The situation came to a head in the 19th century, when the question forced itself on everyone of what reference was light speed in the vacuum of outer space to be taken with respect to. This came out of the treatment of the electromagnetic field that had been formalized by Maxwell.

Maxwell was like Newton, but in the opposite direction. He staunchly supported Galileo's supposition of boost-invariance, but stuck a reference speed in his equations, going out of his way to show that his equations (with the reference speed) were boost-invariant, but then turning around to have it both ways by saying that some "deeper theory" would account for where that universal reference for wave motion in a vacuum came from. So, before Einstein, one made a distinction between the "stationary" and "moving" versions of the equations Maxwell wrote down, and it was only Einstein who showed that the "stationary" version held in all frames of reference.

Einstein resolved the issue by amending Galileo's boost transform in such a way that instead of the infinite speed of being everywhere at once being an absolute speed (as it is, under the Galilean transform) a finite speed would actually be the absolute speed; and that is the speed at which light moves in a vacuum. The cost of doing this is that the speed of being simultaneous (i.e. simultaneity) would be relative. Infinite speed in one frame would be seen as a speed of $c^2/v$ is another frame moving at a speed of $v$ with respect to the first frame, where $c$ is light speed in a vacuum.

(It was only at that point, where the secret marriage of space and time that Galileo had actually de facto ordained was staring everybody in the face, with the amending of Galileo's boost transform, that it finally became clear that spatial and temporal geometry had been wedded by the inclusion of the boost transforms alongside the Euclidean transforms and time transform. But the real minister of that wedding was Galileo back in the 1600's, not those in the 1900's belatedly calling it out.)

For the Galilean transforms, you have the ability to separate out the boosts as a subset of transforms, and treat them separately, since boosts composed with boosts yield boosts. The same applies, separately to spatial translations, time translations and rotations. In each case, the composition of any two transforms of the same type will yield a third transform of the same type.

That's not the case in the transforms that result from Einstein's modification of the Galilean boosts. Two boosts, under Einstein, yield a boost only if they are done in the same or opposite directions and their respective speeds $v$ and $v'$ add up to $(v + v')/(1 + vv'/c^2)$, with the condition that $|v| < c$ and $|v'| < c$ now being required for consistency.

Two boosts along different directions yield the combination of a boost and a small reorientation in direction. That's what lies behind what's called a Thomas Precession https://en.wikipedia.org/wiki/Thomas_precession

A similar situation occurs in the (roughly) spherical geometry of the Earth. The result of two translations on the Earth's surface will yield the combination of a translation and a rotation. In fact, if you do three translations, with the third ending up where you started at, the result will generally be a pure rotation. For instance, take a grid on the equator at 90 longitude west with X pointing east, Y north, translate it north to the north pole. X will then point south along the prime meridian and , Y will point south along the 90 degree east longitude. Then move the grid south along 180 degree longitude back to the equator. Then, X will point north and Y will point west. Then move it back to 90 degrees west longitude: X now points north, Y points west, and the grid has rotated 90 degrees counter-clockwise as a result of the three translations. Similarly, the 6 mile by 6 mile township grids in the old Northwest Territory of the United States have regular defects. A translation, say, 24 miles north then 6 miles east will be off (by a few dozen yards) from the result of a translation 6 miles east first, then 24 miles north, and will be slightly rotated. So, the grid has kinks in it.

If space, itself, had a uniform positive curvature everywhere in the cosmos (as it would in some versions of the Big Bang model) then a similar thing would apply to spatial translations in three dimensions, and the composition of two spatial translations would be a combination of a third spatial translation and a small reorientation.