User:Willow/Cleonis Symmetries

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Contents 1 Symmetries, metrics and transitions between theories 1.1 broadening concepts 1.2 Newton's laws of motion as symmetry principles 1.3 Two transitions and what they have in common 1.4 Overview: the key concepts

[edit] Symmetries, metrics and transitions between theories

[edit] broadening concepts

In transiting from theory to theory, very often concepts are generalized to fit a broader scheme, such as generalizing the concept of simultaneity, and the generalized concept of inertial motion for curved spacetime. I believe the very point of general relativity is the way the concept of inertia has been broadened.

To me the expressions 'theory of minkowski spacetime' and 'theory of inertia' are very much overlapping concepts; whenever I talk about properties of inertia I'm in the same breath talking about properties of spacetime. Our physics theories are in fact structured around inertia (inertia in a geleralized sense), and the surprising thing is that this fact is seldom stated explicitly.

You endorse the point of view that in the transition to GTR (general theory of relativity) there was a shift from assuming that the euclidean metric will be confirmed experimentally to using the more general Riemannian geometry. Well, that point of view is right up my alley. ;-)

[edit] Newton's laws of motion as symmetry principles

Newton's three laws of motion together suffice for a comprehensive theory of motion. From my point of view 'theory of motion' and 'theory of inertia' are one and the same thing. Each of Newton's three laws of motion can be regarded as describing a property of inertia.

A prior assumption of Newton was the physical assumption that for space the euclidean metric holds good. Also, Newton's laws rely on linear transformations (such as the linear galilean transformations) between inertial frames.

We can recognize Newton's first law as asserting a symmetry: when no force is exerted objects move along the straight lines of euclidean geometry, covering equal distances in equal intervals of time. Similarly there is the symmetry under spatial rotation that is stated by the Kepler law: 'equal areas are swept out in equal intervals of time'. Kepler's area law captures that space and time are in a certain relation to each other. In the Principia Newton employed Kepler's area law ingeniously to express passage of time geometrically. In all, Newton's first law asserts uniformity of space, and a certain relation of space and time (inertial motion covers equal distances in equal intervals of time.)

Newton's second law is explicitly about inertia: change of velocity is proportional to the exerted force. And of course, since velocity does not enter the second law, the second law serves to assert relativity of inertial motion, which is a symmetry principle.

Newton's third law can again be recognized as asserting a symmetry: isotropy of inertia. In the principia, the fourth corrollary to the third law says: "The common centre of gravity of two or more bodies does not alter its state of motion or rest by the actions of the bodies among themselves [...]" When two objects exchange momentum the motion of the common centre of mass follows the first law; inertia is the same at all places and in all directions. Conversely, if space-and-time would not be isotropic (equivalently: if inertia would not be isotropic) then the third law would not hold good.

(By the way, I think Newton's three laws can be condensed to two laws without loss of physical content. However, that's another subject.)

[edit] Two transitions and what they have in common

There has been a succession of three significant theories of space and time:
- Newtonian dynamics
- STR (the special theory of relativity)
- GTR (the general theory of relativity)

In the transition from newtonian dynamics to STR the Euclidean metric was replaced with the Minkowski metric. Other than that the symmetries that are asserted by Newton's laws carried over to STR.

In newtonian dynamics all that can be attributed to space(time) is inertia. For the rest newtonian space is pretty bland and passive; the path you take in traveling from one point to another does not matter. By comparison, Minkowski spacetime is a very intrusive spacetime: if you travel along a roundabout path, then less proper time elapses than if you travel along the spatially shortest path - more mileage, less proper time. Once a physicist accepts the Minkowski spacetime metric, he/she realises that a notion of space as empty nothingness is out the window. Special relativity hammers in that spacetime is an active participant in the physics taking place.

The Euclidean metric and the Minkowski metric have the following in common: both are immutable. That is, spacetime is regarded as immutable. The transition from STR to GTR was that according to GTR spacetime is actually subject to influence, deforming in accordance to physical law. In GTR there is a reciprocity: spacetime is acting upon inertial mass and inertial mass is acting upon spacetime. The reason that spacetime itself can be the mediator of gravitational interaction is that relativistic spacetime is so intrusive; by nature relativistic spacetime is acting upon inertial mass.

There's an interesting parallel, I think, in the history of the Kepler problem in newtonian dynamics and the kepler problem in GTR

In newtonian dynamics the Kepler problem is solved by finding two worldlines: the motion of the Sun and the motion of the planet around their common center of mass. That problem is hard to solve, almost insurmountable in Newton's time, for just about everything is in flux. The speed of the planet is not constant, the distance to the Sun is not constant, the gravitational potential is a function of distance to the Sun; all variables of the problem are a function of each other! In order to deal with that level of mutual dependence, Newton developed a new branch of mathematics: differential calculus.
Hard as the problem is, there is a factor that (according to newtonian dynamics) is uniform and homogenous: inertia. (Equivalently: space-and-time is uniform and homogenous.) In newtonian dynamics the solution to the Kepler problem is given as motion described by a function of spatial coordinates and time coordinate.

To solve the Kepler problem in GTR, not just two worldlines need to be found, in GTR the very spacetime metric is a variable of the problem! In order to have a coordinate system to represent the motion in you need a metric, but it's the spacetime metric as a function of the other variables that you seek to solve, along with the other variables. To deal with that, a new branch of mathematics had to be developed. As you know, that new branch of mathematics is called differential geometry.

While STR asserts the same symmetry as Newton's first law does, GTR replaces Newton's first law with a generalized version, usually formulated as an action principle: objects in inertial motion move along a path in (curved) spacetime that maximizes lapse of proper time. As with all generalized versions, in the limit of negligable relativistic effects it converges to the newtonian case.

While newtonian dynamics features a separate theory of inertia and theory of gravitation, GTR is a single, unified theory for inertia and gravitation. A single physical entity, the spacetime as described by the Einstein field equations, gives rise to both inertial and gravitational effects. This unification is an axiom. Given that unification-axiom, the equivalence principle follows as a theorem.

[edit] Overview: the key concepts

A useful divide between theories of space-and-time, I think, is between theories that assume that spacetime is immutable, and a theory in which spacetime not only acts upon inertial mass but is also being acted upon by inertial mass. The general theory of relativity is more general than the special theory in the sense that the general theory describes that inertial mass acts upon spacetime, inducing curvature, whereas this aspect is not present in special relativity.

The reason that spacetime itself can be the mediator of gravitational interaction is that by nature relativistic spacetime is acting upon inertial mass.

In transiting from theory to theory, very often concepts are generalized to fit a broader scheme, such as generalizing the concept of simultaneity, and the generalized concept of inertial motion for curved spacetime. I believe the point of general relativity is the way the concept of inertia has been broadened.

I think that for fundamental understanding the best thing is to concentrate on what the Newtonian-to-STR transition and the STR-to-GTR transition have in common. What changed in those transitions was the spacetime metric. Arranged in a table:

Theory	metric
Newtonian dynamics	Euclidean metric + uniform progress of time
Special relativity	Minkowski metric
General relativity	dynamic (-,-,-,+) signature Riemannian metric

The socalled fictitious forces 'centrifugal force' and 'coriolis force' do not have a physical existence, in the same sense as coordinate systems do not have a physical existence. The statement 'gravity acts like a fictitious force such as the centrifugal force or the Coriolis force' compares something that exists with something that doesn't exist, so that's just gobbledygook. --Cleonis | Talk 20:17, 17 October 2007 (UTC)