[Alinta (OP)]

Hi guys,

Ok let's start with the hypothetical analogy of a plane going around the planet - it goes in what it thinks is a straight line, but actually that line is curved.

However if that plane actually flew from a point on earth to another point in the universe, is that line also curved?

I guess another way of asking is: I can see how space-time curvature and the imaginary 'dents' created in the space-time blanket by all these big spheres (planets, stars etc) can make curves appear straight, but I am confused by the fact that the objects of reference in all the space-time things that I am reading are basically spherical in shape. (planets, stars, etc). So is this curve the same for all/other situations? If the plane leaves Earth's gravity and heads towards a different source of gravity, or is pulled slightly in all different ways between here and a distant star, from many different 'dents', what would the line from point A to point B look like then? Presumably it still appears straight to the pilot in the plane, but would the actual line still be curved? Would it be curved in all different directions, depending on the nearest masses creating gravity, or just one gradual curve, like the plane going around the planet? Or, similarly, is space-time still curved on a planet or large mass the shape of a cube?

[yor_on]

Two ways. One is a acceleration
The other is called 'uniform motion'

Uniform motion can also be called a geodesic. a geodesic is the path of 'least' or 'no' resistance taken if you think of gravity as something restricting a motion to certain paths. If we go by measuring locally there is no 'resistance' at all in a uniform motion which makes a 'geodesic' a path of no resistance. But as you say, all mass have a gravitational influence, and shouldn't there be some 'resistance' as they act on each other, also having a effect on the path defined by the geodesic?

I'm not sure there, if it has it's too small to be measurable.

But a acceleration breaks any geodesic. So using that you can get a euclidean straight path, what we normally think of as a straight line, but only by expending energy. and if we think of it in form of 'resistance', any acceleration meets a resistance (inertia) and that's why it expends energy. But as fast as you stop accelerating you're back into a geodesic.

[syhprum]

I am puzzled by this talk of comparing geodesics with straight lines surely in space there is no such thing as a straight line geodesics are the only thing, when we try to represent a geodesic on a two dimensional surface it looks curved but space is not two dimensional.

[Pmb]

Ok let's start with the hypothetical analogy of a plane going around the planet - it goes in what it thinks is a straight line, but actually that line is curved.

No. That is incorrect. A plane does not think its going in a straight line. It thinks its changing direction relative to the distant stars. If it were to go in a straight line then it would almost follow a path parallel to a photon.

I think there is some confusion in this thread. Let’s dispose with the notion of a “straight” line in a curved space. Let us, instead, speak of a path of extremal “interval.” I use the term “interval” here only because I don’t know of another term of the integral of the spacetime interval. Calling it :”length” seems to get people into trouble because the assume that this “length” divided by a time will equal some sort of “spacetime velocity” and such a notion is nonsense.  What I’ve just described is referred to as a “geodesic.” Some people think of the geodesic as the “straightest possible line in a curved space.”

I used the term “extremal” because the interval need not be a minimum. Think of a cylinder. The surface of a cylinder has zero intrinsic curvature. A geodesic on such a surface could be a helix going around the cylinder. If you were to cut the cylinder along its length and lay if out flat there would be a bunch of lines on it. Two points on the surface along its length can be connected by an infinite number of helixes of a straight line. All of them will be geodesics. The line going down parallel to its axis will be a minimal length while there is an infinite number of helixes of varying lengths.

A particle subjected to zero non-inertial forces moves in a geodesic in spacetime. However the interval is not ds^2 = dx^2 + dy^2 + dz^2 like regular geometry but is ds^2 = (cdt)^2 - dx^2 - dy^2 - dz^2 (in locally Lorentzian coordinates). It is the integral of this quantity that must be extremal.

[evan_au]

We often use the example of someone sailing or flying over the Earth, because we understand that their motion is constrained to the curved 2-dimensional surface of the Earth - this 2-dimensional surface curves "into" a 3rd dimension.

However, it is much harder to get a mental image of what a curved 3-dimensional space looks like: what does it curve "into"?

One way of imagining it is the traditional formula for the Circumference of a circle in Euclid's geometry: C=πD

• Where the Diameter D is related to the Circumference C by the constant Pi≈3.14159...
• ...But if you measure the Diameter and Circumference of different-sized circles drawn on the surface of the Earth, you will find that the ratio is different from the traditional value of Pi. This tells you that the Earth is not a "Euclidean" surface.
• Similarly, if you measure the Diameter and Circumference of different-sized circles drawn near a star or a black hole, you will find that the ratio is slightly different from the traditional value of Pi. This tells you that 3-dimensional space is not a "Euclidean" space.
• A similar demonstration can be done with the sum of angles in a triangle, which always add to 180° in Euclid's geometry, but don't always in "real" 2D & 3D geometry.

Black holes are a bit hard to reach, but you can see some of these distortions of space in the way light is bent when it passes near a strong gravitational field, like a dense cluster of galaxies. This is called gravitational lensing. Light travels in straight(ish) lines through space, but the distortions of space can make distant objects appear spread out or to appear in multiple places slightly separated in the sky. Some astronomers are trying to "undo" these distortions of distant objects (due to multiple "dents" in space) to create an image of the invisible closer object.

See http://en.wikipedia.org/wiki/Gravitational_lensing