Gravity is a Force !

[studiot]

 

 

Commander

Of course that was an yes !

 

 

 

 

OK so geodesics. They have an interesting story.

 

Wikipedia does a fair job of summarising facts. I have picked out some salient points and numbered them.

 

http://en.wikipedia.org/wiki/Geodesic

 

 

 

Wikipedia

In mathematics, particularly differential geometry,

 

(1)a geodesic is a generalization of the notion of a "straight line" to "curved spaces".

 

In the presence of an affine connection,

 

(2)a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it.

 

If this connection is the Levi-Civita connection induced by a Riemannian metric, then the geodesics are (locally) the shortest path between points in the space.

 

(3)The term "geodesic" comes from geodesy, the science of measuring the size and shape of Earth; in the original sense, a geodesic was the shortest route between two points on the Earth's surface, namely, a segment of a great circle.

 

The term has been generalized to include measurements in much more general mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph.

Geodesics are of particular importance in general relativity. Geodesics in general relativity describe the motion of inertial test particles.

 

(4)The shortest path between two points in a curved space can be found by writing the equation for the length of a curve (a function f from an open interval of R to the manifold), and then minimizing this length using the calculus of variations.

 

This has some minor technical problems, because there is an infinite dimensional space of different ways to parameterize the shortest path. It is simpler to demand not only that the curve locally minimize length but also that it is parameterized "with (5)constant velocity", meaning that the distance from f(s) to f(t) along the geodesic is proportional to |s−t|. Equivalently, a different quantity may be defined, termed the energy of the curve; minimizing the energy leads to the same equations for a geodesic (here "constant velocity" is a consequence of minimisation). Intuitively, one can understand this second formulation by noting that an elastic band stretched between two points will contract its length, and in so doing will minimize its energy. The resulting shape of the band is a geodesic.

(6)In Riemannian geometry geodesics are not the same as "shortest curves" between two points, though the two concepts are closely related. The difference is that geodesics are only locally the shortest distance between points, and are parameterized with "constant velocity". Going the "long way round" on a great circle between two points on a sphere is a geodesic but not the shortest path between the points. The map t → t² from the unit interval to itself gives the shortest path between 0 and 1, but is not a geodesic because the velocity of the corresponding motion of a point is not constant.

Geodesics are commonly seen in the study of Riemannian geometry and more generally metric geometry. In general relativity, geodesics describe the motion of point particles under the influence of gravity alone. In particular, the path taken by a falling rock, an orbiting satellite, or the shape of a planetary orbit are all geodesics in curved space-time. More generally, the topic of sub-Riemannian geometry deals with the paths that objects may take when they are not free, and their movement is constrained in various ways.

 

 

 

 

Unfortunately, Wiki gets the history wrong by nearly 2000 years in point (note 3).

 

Geodesic is an Ancient Greek word ‘geodaisia’ which means ‘divides the Earth’ from two Ancient Greek Words geos- the Earth and daiesthai - to divide.

To understand this you need to understand that in 2000+ years ago Science, Philosophy and Religion were inextricably linked. The Greeks of that time regarded ‘perfect’ shapes is proper and real and many cultures had a Royal Road that could only be traversed by the ruler-cum-deity. Since to them (some at least) a sphere was the perfect 3D shape, the Earth was a sphere.

This road did not deviate to right or left from its path (kept straight on). So if one followed it one would eventually come back to one’s starting point and one’s path would ‘divide the Earth equally’.

This accords with the notion in (note 2).

 

They were, of course, talking about great circles on the globe.

 

Moving on nearly 2000 years cartographers realised that the minor arc of a great circle on a sphere is the shortest distance between two points.

Unfortunately they also realised that the Earth is not quite a perfect sphere, and the modern idea of a geodesic as the shortest distance on the real shape of the earth was born.

So the mathematicians of the time got hold of the issue (which is where the Wiki history starts) and they realised that geodesics had other properties, besides shortest distance.

 

(Note 1) shows that the proved that geodesics on a plane are straight lines and that lead to the mathematics of developable surfaces and ‘ruled’ lines.

 

(Note 6) was the culmination of 100 years of post renaissance maths development, primarily in Europe where the geodesic as the shortest line between two points on a surface dominated.

 

This period, to the mid 19th century, also saw the development of much of the apparatus of modern maths, in particular the idea of manifolds (note4) instead of surfaces.

 

All these were variations on the cartographic idea of a geodesic, where the scale is even on all axes.

 

This brings us to (note5) which introduces the next instalment of the story and provides the link to geodesics as minimisers (or maximisers) of other expressions besides distance.

Edited February 2, 2015 by studiot

[MigL]

This is all fine and dandy.

We now have various explanations of the workings, history and etymology of geodesics.

 

But I believe the OP was along the lines of ' is gravity a force ? '.

 

I see very little discussion as to whether gravity is an 'action at a distance' force mediated by force carrier bosons ( gravitons ? ), or, a particular arrangement of space-time such that mass ( and energy ) is constrained to move along the resultant geodesics.

[ajb]

I see very little discussion as to whether gravity is an 'action at a distance' force mediated by force carrier bosons ( gravitons ? ), or, a particular arrangement of space-time such that mass ( and energy ) is constrained to move along the resultant geodesics.

The two are not competing ideas, but rather complimentary ones, just as we are not discussing if the electromagnetic force is due to photons or arrangements of the electromagnetic field due to the presence of charges. Loosely, gravitons correspond to quantised gravitational waves just as photons correspond to quantised electromagnetic waves.

[Reaper79]

Regarding the OP

 

I often had similar musings, physicists says, gravity sucks, a nice turn of phrase. But I would lie awake wondering if gravity was pushing or pulling me,

if space itself could somehow determine the effect of gravity depending on how massive the object was, and if *space* was pushing us down or

if the Earth was pulling us down.

 

It's an interesting enough concept, but I think all the thinking has been done on this topic already.