Beginner Help on Minkowski Space (and others)

[dekoi]

Please help me in any way you can. I'm a beginner physics student (entering undergraduate engineering) and currently self-taught.

1.) Minkowski space is illustrated in one of my books as a cone, with it's apex as the "origin", the side as the "light cone", the bend curvature inside the cone as the "hypoerbolic space" and a point inside the cone as "a point in spacetime". Please explain this to me.

2.) How does gravity make space non-Euclidean? Is it because gravity bends space, which causes non-Euclidean assumptions (e.g. angles in a triangle add up to 180 degrees, and parallel lines are possible) to be false?

3.) How effective is gravitational lensing in distorting our perception of the world? Would it be proper to assume that the 'actual' position of a still object on Earth deviates a fraction of a millimeter (or smaller) from our human perception due to this phenomenon?

4.) Why excactly don't the laws of Euclidean geometry hold in a uniformly rotating system? A text states that this occurs because space warps due to the contraction of the circumference. Why? Does this have anything to do with the centrifugal force? If so, doesn't the centrifugal force 'push away' objects from their original path and thus cause an expansion of the circumference?

5.) "Imagine a circle spinning in space. According to special relativity, the boundary of the disk contracted as it spun. There was a force acting on the circle at the boundary -- the centrifugal force -- and its action was analogous to that of a gravitational force. But the same contraction that affected the outer circle left the diameter unchanged. Thus...the ration of the circle to the diameter was no longer pi."
i. Once again, the centrifugal force is said to contract the boundary (like the gravitational force). I was under the impression that the centrifugal force pushes objects away from centripetal motion - -and it is actually centripetal force which pushes them towards the orbit's center. Also, why does the diameter remain unchanged?

 

[robphy]

1.) Minkowski space is illustrated in one of my books as a cone, with it's apex as the "origin", the side as the "light cone", the bend curvature inside the cone as the "hypoerbolic space" and a point inside the cone as "a point in spacetime". Please explain this to me.

The cone, called the Light Cone, is not Minkowski spacetime... it is a subset of Minkowski spacetime (with dimensionality one less than Minkowski spacetime). It is [in one interpretation] the set of events that is causally-accessible to or from the vertex with a light ray. Its interior contains the set of events that is "causally-accessible with a massive particle (used as a signal)".. this is sometimes called timelike-accessible. Those outside are not causally-accessible to or from the vertex. Each "point of Minkowski spacetime" is an "event in spacetime". Each event in spacetime has its own Light Cone.

Alternatively, the cone represents sets of analogous directions [in the tangent space]. It is featured because, unlike Euclidean space, Minkowski spacetime is not isotropic. In Minkowski spacetime, there are preferred directions [eigenvectors of Lorentz Transformations] which physically correspond to the invariant speed of light. This divides the set of all possible directions from the vertex into three classes: lightlike (or null), timelike, and spacelike.

The hyperboloid inside the light cone represents the set of events that are "equidistant" (that is, at a fixed proper-time interval) from the vertex. The unit hyperboloid can also be identified with the "space of timelike 4-velocities" based at the vertex, which has the geometry of a hyperbolic space [of dimensionality one less than that of Minkowski space].



When I have time, I will try to address the other questions

 

[εllipse]

2.) How does gravity make space non-Euclidean? Is it because gravity bends space, which causes non-Euclidean assumptions (e.g. angles in a triangle add up to 180 degrees, and parallel lines are possible) to be false?

The idea for non-Euclidean geometry stems from Euclid's parallel postulate (the fifth postulate in his Elements), which states, "That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles." Commentators as soon as Proclus realized that the parallel postulate wasn't obvious enough to be a postulate. Proclus tried restating the parallel postulate as "a straight line which meets one of two parallels must also meet the other" and then tried to prove it, rather than letting it be assumed. But the parallel postulate could never be proven conclusively, so some mathematicians wondered what goemetries could be constructed if the parallel postulate was dropped. Eventually, Riemann's version of non-Euclidean geometry was picked up by Einstein and used in general relativity. The important thing to note is that space being Euclidean shouldn't be assumed. In your question you make it sound like gravity distorts a truer form of geometry into something messy, when, in fact, Euclid was just blinded by his experiences which seemed to indicate that parallel lines would always remain parallel. But Euclid's assumption wasn't founded on any obvious property of nature, as is proven by the fact that two thousand years before Einstein mathematicians began disputing the parallel postulate.

4.) Why excactly don't the laws of Euclidean geometry hold in a uniformly rotating system? A text states that this occurs because space warps due to the contraction of the circumference. Why?

If you are watching a rotating disc, you have to conclude that due to special relativity the circumference will contract, since the circumference is moving relative to you.

5.) "Imagine a circle spinning in space. According to special relativity, the boundary of the disk contracted as it spun. There was a force acting on the circle at the boundary -- the centrifugal force -- and its action was analogous to that of a gravitational force. But the same contraction that affected the outer circle left the diameter unchanged. Thus...the ration of the circle to the diameter was no longer pi."
i. Once again, the centrifugal force is said to contract the boundary (like the gravitational force). I was under the impression that the centrifugal force pushes objects away from centripetal motion - -and it is actually centripetal force which pushes them towards the orbit's center. Also, why does the diameter remain unchanged?

The important thing to note is that if you are riding along on the rotating disc, you do not have to assume the disc is spinning. It is possible to create a reference frame for the spinning disc (the general principle of relativity), so if you assume the disc is not spinning, the force pulling you away from the edge of the disc will be gravitational, not centrifugal. The diameter won't contract because there is no motion in the direction of the diameter, and Lorentz contractions only contract in the direction of motion.

For a good description of this, see Einstein's Relativity: The Special and General Theory, Chapter 23.

Also, you will probably find this thread helpful: https://www.physicsforums.com/showthread.php?t=84734

Note pervect's posts on some of the problems with creating a reference frame for a spinning disc.