[Widdekind]

I understand, that the "mom-energy" 4-vector is calculated, from quantities, derived from two different reference frames. First, the observer measures a 4-displacement, , between two events, on the trajectory, of some particle, propagating through their laboratory rest frame. Then, that 4-displacement is divided by the proper time, , elapsing between those two events, in the particle's own rest-frame. I.e. "the observer measures the displacement, but the particle reports the time".

The "mom-energy" is the velocity 4-vector, multiplied by the particle's mass. What about 4-acceleration ? I understand that

If the 4-velocity "points" in the direction of the particle's time-axis, i.e. "is the tangent vector to the particle's world-line"; then must the orthogonal 4-acceleration "point" in the direction of the particle's space-axis ?? I observe, that, indeed, in a particle's own rest-frame, the 4-acceleration has no time component, and so is wholly space-like. More generally, in (2+1)D, I suspect, that the 4-acceleration must lie in the "tilted (x'y') plane" in the particle's reference frame.

[DrRocket]

michel123456, on 7 February 2012 - 03:06 PM, said:

Geodesics between the poles is a special case. Only when the 2 points are exactly opposite (antipodes) there are multiple geodesics. Between 2 random points on Earth's surface, there is only one geodesic which is the smallest path.

Nope.

On a sphere there are at least two geodesics between any two points -- each a segment of a great circle passing through the two points. Geodesics, even in the truly Riemannian case, do not necessarily minimize the arc length between two points, they only do so for points that are sufficiently close to one another. In the case of the Lorentzian metric used in general relativity, geodesics actually maximize local arc length.

[Widdekind]

IM Egdall, on 6 February 2012 - 10:36 PM, said:

http://physics.bu.ed...8_Momenergy.pdf

Please ponder "mom-energy", in curved space-time. Particles residing near massive bodies experience gravitational time dilation. Accordingly, for a distant observer, dividing by the particle's proper time, induces a "gravitational gamma factor", . Note, this relation can also be derived, from , using the Schwarzschild metric, on a stationary test charge , which "picks out" the time-time component, of the metric tensor, . If so, then particles residing near massive bodies, are perceived, by remote observers, to have a gravity time dilation; and a gravity gamma-like factor, which increases the effective mass, i.e. "time slows & mass increases, for fast-moving particles; and for particles in gravity 'wells'".

principle of equivalence:

Whether in "flat" space-time, or "curved" space-time, particles "prefer" to propagate along "straightest possible paths", i.e. geodesics, through space-time. Per Newton's first law, each & every deviation, from the "desired" geodesic "free fall" path, must result from forces; and are all experienced equally as accelerations. Is this accurate ?

I understand, that, ultimately, all force-ful interactions arise, from the three fundamental forces, i.e. Strong, Weak, EM, via boson exchange. E.g. standing on earth is, ultimately, an EM interaction, between electrons, via virtual photons; whereas standing on a neutron-star would, hypothetically, be a Strong interaction, via virtual pions. Is that accurate ?

[imatfaal]

IME - that's not the definitions I would have used - take a look at this page on Gravitational Time Dilation

I know wiki can sometimes play a little fast and loose but here is one section

Quote

A common equation used to determine gravitational time dilation is derived from the Schwarzschild metric, which describes spacetime in the vicinity of a non-rotating massive spherically-symmetric object. The equation is:

, where

• t₀ is the proper time between events A and B for a slow-ticking observer within the gravitational field,
• t_f is the coordinate time between events A and B for a fast-ticking observer at an arbitrarily large distance from the massive object (this assumes the fast-ticking observer is using Schwarzschild coordinates, a coordinate system where a clock at infinite distance from the massive sphere would tick at one second per second of coordinate time, while closer clocks would tick at less than that rate),

The way I think of it is that proper time is the time between two events as experienced by the clock that passes through both these events. Coordinate time is more complicated - but in a situation like this is that of a distant theoretical observer

This post has been edited by imatfaal: 8 February 2012 - 12:24 PM

[IM Egdall]

imatfaal, on 8 February 2012 - 12:24 PM, said:

IME - that's not the definitions I would have used - take a look at this page on Gravitational Time Dilation

I gave the definition of coordinate time in special relativity (flat spacetime -- zero gravity). You gave the definition in general relativity (curved spacetime).

Widdekind, on 8 February 2012 - 12:32 AM, said:

I understand, that the "mom-energy" 4-vector is calculated, from quantities, derived from two different reference frames. First, the observer measures a 4-displacement, , between two events, on the trajectory, of some particle, propagating through their laboratory rest frame. Then, that 4-displacement is divided by the proper time, , elapsing between those two events, in the particle's own rest-frame. I.e. "the observer measures the displacement, but the particle reports the time".

In special relativity, momenergy is calculated in a single reference frame. The 4-displacement is measured in that frame, as you indicate. But the proper time is invariant -- it is the same for all (uniformly moving) frames of reference.

This post has been edited by IM Egdall: 8 February 2012 - 03:03 PM

[michel123456]

DrRocket, on 8 February 2012 - 12:49 AM, said:

Nope.

On a sphere there are at least two geodesics between any two points -- each a segment of a great circle passing through the two points. Geodesics, even in the truly Riemannian case, do not necessarily minimize the arc length between two points, they only do so for points that are sufficiently close to one another. In the case of the Lorentzian metric used in general relativity, geodesics actually maximize local arc length.

Why nope?
I suspect a language issue.
I maintain: on a sphere between 2 random points there is only one geodesic which is the smallest path, meaning that any other geodesic has a longer path. When the longest path is equal to the smallest, then the 2 points are at the antipodes, and only in this case there are multiple path of the same length.
That was in response to your statement that

Quote

But there can be more that one path that exhibits that shortest distance. Consider lines of longitude on the globe. They are all geodesics between the poles.

I don't think such a configuration can be found easily in outer space: the Universe has no poles and random points in space cannot be found at the antipodes of the universe.

This post has been edited by michel123456: 8 February 2012 - 06:55 PM

[StringJunky]

michel123456, on 8 February 2012 - 06:06 PM, said:

Why nope?
I suspect a language issue.

It's just a more emphatic version of "No"....I suspect he's confident it's wrong.

[imatfaal]

My school level maths agrees with Michel. Assuming a perfect sphere and not the squidged shape of earth (and possibly with that too), then the only two points with more than one geodesic - ie that there are multiple shortest routes between are antipodes. Any point that are not antipodes will have one and only one shortest route.

reasoning:
The shortest route is a section of a great circle.
A great circle is the intersection of the surface of the sphere and a plane which passes through the centre of a sphere.
There can be only one plane which passes through three points as long as those points are not in a line.
The points cannot be in a line (start of journey, end of journey, centre of earth) unless the start and finish are antipodal

[Santalum]

Anilkumar, on 7 February 2012 - 04:35 AM, said:

In Euclidian geometry; the shortest path between any two points is the straight line.

In non-Euclidian geometry; the shortest path between any two points is the Geodesic.

In either case, there cannot be more than one shortest distance.

A geodesic is an arc according to this: http://en.wikipedia.org/wiki/Geodesic

And the shortest distance between two points in 4D spacetime could be a wormhole........equivalent to boring through the earth from one point on the surface to another.

Correct?

[michel123456]

imatfaal, on 9 February 2012 - 02:03 PM, said:

My school level maths (...)

You are too modest.

[Widdekind]

IM Egdall, on 8 February 2012 - 03:13 PM, said:

In special relativity, momenergy is calculated in a single reference frame. The 4-displacement is measured in that frame, as you indicate. But the proper time is invariant -- it is the same for all (uniformly moving) frames of reference.

"mom-energy" = [ "space-time 4-displacement" between events A,B ] divided by [ "proper time" ]

I understand you to be telling me, that "proper time" -- which, in my mind, is specific to that reference frame "threading through" both events A,B -- is better thought of, as the "invariant interval .

In my mind,

"universally invariant interval" dS between events A,B

is distinct from

"proper time"

which is limited, to that special reference frame, in which the whole entire invariant interval dS is "perceived" to be "elapsed time" dS = dt_p, i.e. in every other reference frame, the invariant interval, although constant, is composed of a time part and a space part, whose difference remains invariant.

In my wordage, I understand you to be telling me, that

"momenergy" = "4-displacement" / "invariant interval dS"

Intuitively, that seems a better definition, on the 'grounds' that the invariant interval dS is "directly accessible" to every reference frame, without any naive notions, of the perceived-to-be-moving particle somehow "reporting its proper time". To my mind, distinguishing "invariant interval" (same for all observers) from "proper time" (specific to a special reference frame) helps clarify this concept.

[IM Egdall]

Widdekind, on 9 February 2012 - 09:27 PM, said:

"mom-energy" = [ "space-time 4-displacement" between events A,B ] divided by [ "proper time" ]

Intuitively, that seems a better definition, on the 'grounds' that the invariant interval dS is "directly accessible" to every reference frame, without any naive notions, of the perceived-to-be-moving particle somehow "reporting its proper time". To my mind, distinguishing "invariant interval" (same for all observers) from "proper time" (specific to a special reference frame) helps clarify this concept.

I agree.

[DrRocket]

michel123456, on 8 February 2012 - 06:06 PM, said:

I don't think such a configuration can be found easily in outer space: the Universe has no poles and random points in space cannot be found at the antipodes of the universe.

What you think or believe is irrelevant. It is what you can support with evidence or prove from established theory that counts.

You might consider the fact that one of Einstein's preferred models for the universe modeled space as a 3-sphnere.

Santalum, on 9 February 2012 - 03:09 PM, said:

A geodesic is an arc according to this: http://en.wikipedia.org/wiki/Geodesic

And the shortest distance between two points in 4D spacetime could be a wormhole........equivalent to boring through the earth from one point on the surface to another.

Correct?

Wiki strikes again. The second definition is the correct one. On a Riemannian manifold there is always an affine connection, the Levi-Civita connection that produces the metric.

A geodesic is only length minimizing locally -- in a sufficiently neighborhood of a point. This is not a good definition, but rather is a consequence of the proper definition noted above.

On manifolds that are not geodesically complete there are points that are not connected by geodesics. On other manifolds there may be points connected by more than one, and perhaps infinitely many geodesics.

I would sugges that some people in this thread (not particularly you) might want to actually learn some differential geoometry prior to opening their mouths and making fools of themselves.

This post has been edited by DrRocket: 10 February 2012 - 02:05 AM

[Santalum]

DrRocket, on 10 February 2012 - 02:04 AM, said:

What you think or believe is irrelevant. It is what you can support with evidence or prove from established theory that counts.

You might consider the fact that one of Einstein's preferred models for the universe modeled space as a 3-sphnere.



Wiki strikes again. The second definition is the correct one. On a Riemannian manifold there is always an affine connection, the Levi-Civita connection that produces the metric.

A geodesic is only length minimizing locally -- in a sufficiently neighborhood of a point. This is not a good definition, but rather is a consequence of the proper definition noted above.

On manifolds that are not geodesically complete there are points that are not connected by geodesics. On other manifolds there may be points connected by more than one, and perhaps infinitely many geodesics.

I would sugges that some people in this thread (not particularly you) might want to actually learn some differential geoometry prior to opening their mouths and making fools of themselves.

Can I take it that, on a 2D curved surface a geodesic = an arc but that this does not hold for higher dimensional curves?

I suppose it should be obvious that an arc is specific to a 2D curved surace but that a geodesic is more generalised and covers higher dimensions.

Perhaps Wikipedia should explicitly state this if it is the case because it currently appears that geodesic = arc may be assumed by some readers.

This post has been edited by Santalum: 10 February 2012 - 02:52 AM

[DrRocket]

Santalum, on 10 February 2012 - 02:49 AM, said:

Can I take it that, on a 2D curved surface a geodesic = an arc but that this does not hold for higher dimensional curves?

no

Santalum said:

I suppose it should be obvious that an arc is specific to a 2D curved surace but that a geodesic is more generalised and covers higher dimensions.

no

Santalum said:

Perhaps Wikipedia should explicitly state this if it is the case because it currently appears that geodesic = arc may be assumed by some readers.

A geodesic is curve. The dimension of the manifold is irrelevant.

A geodesic is a smooth curve with the additional costraint that the family of tangent vectors along the curve is parallel along the curve.

There is no simple explanation of this. You need to read a book on differential geometry.

[Santalum]

DrRocket, on 10 February 2012 - 05:13 AM, said:

A geodesic is a smooth curve with the additional costraint that the family of tangent vectors along the curve is parallel along the curve. There is no simple explanation of this. You need to read a book on differential geometry.

If you take a given latitude on the earth's surface and draw tangents all the way along that latitude line then they would all converge some where above the north or south pole and form a cone. If you take a given longnitude on the earth's surface and draw tangents along it then none of them will be parellel. So I don't understand how you can have parallel tangents........or have I some how misunderstood what you meant by this? Can you direct me to some sort of diagram that will enable me to visualise what you mean by this? Here is another site that describes a geodesic as a great arc: http://gregegan.cust...warzschild.html

Quote

On the surface of the Earth, the closest thing we have to a straight line is known as a great circle, or geodesic. The geodesic between two cities is the route that planes would fly in a perfect world (ignoring various logistical complications), because it always comprises the shortest total distance. In a three-dimensional view of the Earth, the geodesic that joins city A with city B is an arc of the circle with a radius as large as that of the Earth itself, that passes through both cities, and whose centre lies at the centre of the Earth. But we don't need to take that Earth-from-space view; we could equally well define the geodesic from A to B by requiring that everywhere along its length, it acts locally like the straightest possible line: as we walk along it, we won't detect any swerving to the left or right.

As I understand from this the distinction between a great arc and an arc is that an arc is a curved line segment with tangential line segments along its length but a great arc is a curved line segment within a curved surface which has multiple tangential planes along its length. <BR><BR>

If you draw lines from two points on the earth's surface and the center of the earth then the line joining those two points on the earth's surface is an arc of a ccircular slice of the earth. So surely in 2D curved surfaces the distinction between an arc and a geodesic is some what blurred.

<BR><BR>I obviously can't picture this sort of scenario in 4D spacetime so, without learning the mathematics, I will just have to take your word for it that there is a clear distinction between an arc and a geodesic in that case.

This post has been edited by Santalum: 10 February 2012 - 10:08 AM

[michel123456]

DrRocket, on 10 February 2012 - 02:04 AM, said:

(...)
You might consider the fact that one of Einstein's preferred models for the universe modeled space as a 3-sphnere.
(...)

What is a 3-sphnere?

[imatfaal]

michel123456, on 10 February 2012 - 12:54 PM, said:

What is a 3-sphnere?

The surface of a football or similar (known in maths as a ball) is a sphere - also known as a 2-sphere. It is the locus of all points in 3d space equidistant from a single point (the single point being the centre of the sphere).

A 3-sphere is the equivalent shape with an extra dimension. A n-sphere is the locus of all points that are the same distant but in every direction from a chosen point in n+1 space - ie a 2 sphere is all the points in space that are exactly the same distance in 3 dimensions from your given point. it is called a 2 sphere because is it just the surface - in laymans terms if you are on the 2 sphere there are only two directions in which you can move (ie north/south and clockwise/counter-clockwise) even though the construct is in 3d space.

I am still curious to see anyone give more than one shortest route between london and new york - I still cannot see how a sphere/2-sphere can have more than one shortest route (except between antipodes)

[DrRocket]

Santalum, on 10 February 2012 - 08:20 AM, said:

If you take a given latitude on the earth's surface and draw tangents all the way along that latitude line then they would all converge some where above the north or south pole and form a cone.

The only latitude that is a geodesic is the equator.

Santalum, on 10 February 2012 - 08:20 AM, said:

. So surely in 2D curved surfaces the distinction between an arc and a geodesic is some what blurred.

Wrong.

All geodesics are curves (arcs). Very few arcs are geodesics.

michel123456, on 10 February 2012 - 12:54 PM, said:

What is a 3-sphnere?

It is anything topologically equivalent to the set of all points that are some fixed distance from the origin in 4-space.

A 2-sphere is anything that is topologically equivalent to the set of all points that are a fixed distance from the origin in 3-space -- like the surface of a globe.\

Dimension refers to the object itself, not the dimension of some space in which you might find it embedded. So a 2-sphere is two dimensional because in small local patches it "looks like" a plane.

Santalum, on 10 February 2012 - 08:20 AM, said:

As I understand from this the distinction between a great arc and an arc is that an arc is a curved line segment with tangential line segments along its length but a great arc is a curved line segment within a curved surface which has multiple tangential planes along its length.

Apparently you don't understand it at all. This makes no sense.

Go read a book on topology and geometry. The book by Singer and Thorpe would be a good place to start.

[Santalum]

DrRocket, on 11 February 2012 - 01:45 AM, said:

Apparently you don't understand it at all. This makes no sense.

Go read a book on topology and geometry. The book by Singer and Thorpe would be a good place to start.

Yes apparently I don't. I am struggling to comprehend what you mean.

But now you have me absolutely intrigued even though I have never had any particular interest in maths.

I don't suppose there is a website as good as your book that I could read? Anyone?