Interpretation of spacetime diagrams.

[matheinste]

Hello all

I have been at ease with spacetime diagrams of the usual sort where the y and z dimensions are suppressed until I came across this passage from the following book.

Problems of Space and Time. A reader edited and with an introduction by J.J.C.Smart. Page 293 of the chapter entitled The Four Dimensional World by Moritz Schlick. Reprinted from Chapter 7 of The Philosophy of Nature 1949.

-----The world-lines describe the motion of particles; but they must not be mistaken for the tracks of these particles. One may not, for example, that a point traverses its world-line; or that the three dimensional section which represents the momentary state of the actual present, wanders along the time axis through the four dimensional world. For a wandering of this kind would have to take place in time; and time is already represented within the model and cannot be introduced again from outside.------

My understanding was that a point on a world-line represented the position of a particle in space at a certain time, a sort of plot of position, or track, against time. So I am either mistaken or I fail to understand the significance of the above passage, or both. Can anyone expand on this.

Matheinste.

 

[tiny-tim]

Hello matheinste!

Yes, I think that book is a little confusing …

a world-line is in 4D, so it can't be a track, which would be in 3D.

But the projection of the 4D world-line into the spatial 3D is the track of the particle,

and as you go "up" the 4D graph, in the t-increasing direction, the projection certainly traverses the track of the particle, and I see nothing wrong with saying that increasing t also takes you along the world-line.

(For example, the world-line of the Earth, relative to a fixed Sun, is a 4D helix, whose 3D projection is an ellipse, and increasing t along the helix does genuinely show you where the Earth is at each time)

 

[matheinste]

Thanks tiny-tim.

That's helpful. It's roughly what I thought. Perhaps my mistake is thinking of a worldline as a curve in three dimensional space parametized by time.

Matheinste.

 

[Naty1]

A point DOES traverse it's world line, but that is not the physical track we observe here in three D space as noted above. The world line of a constant velocity particle is a straight line (which we happen to observe here); but an accelerating particle is a curved world line ( which we would normally observe as a straight line acceleration) and as noted above an object in constant circular motion (like an electron orbiting a nucleus) is a helix (corkscrew) world line...They are two equally valid descriptions from different perspectives...

 

[matheinste]

I looked in the following book to get the explanation below which makes it clearer to me.
From Schlick, Space and Time in Contemporary Physics-Theory of Relativity and Gravitation, Dover (2005). Page 51.

----let us suppose a point to move in any way in a plane (that of may be chosen). It describes some curve in this plane. If we draw this curve, we can, by looking at it, get an impression of the shape of its path, but not of any other data of its motion, e.g., the velocity which it has at different points of its path, or the time at which it passes through these points. But if we add time, as a third coordinate, the same motion will be represented by a three dimensional curve, the form of which immediately gives us information about the character of the motion; for we can recognize from it which belongs to any point of the path, and we can also read off the velocity at any moment from the inclination to the plane. We shall follow Minkowski by appropriately calling this curve the world-line of the point. A circular motion in the plane would be represented by a helical world-line in the manifold. This trajectory of the point only arbitrarily expresses, as it were, one aspect of its motion, viz. the projection of the tree dimensional world-line on the plane. Now, if the motion of the point itself takes place in three dimensional space, we obtain for its world-line a curve in the four dimensional manifold of the , and from this line all characteristics of the motion of the point can be studied with the greatest of ease. The path of the point in space is the projection of the world-line on the manifold of the , and thus gives an arbitrary and one-sided view of a few properties only of the motion: whereas the world-line expresses them all in their entirety.----

Matheinste.

 

[atyy]

From Schlick, Space and Time in Contemporary Physics-Theory of Relativity and Gravitation, Dover (2005). Page 51.

We shall follow Minkowski by appropriately calling this curve the world-line of the point.

Cool! I always wondered who invented the term.

 

[matheinste]

Hello all

I'm afraid that in #5 mathematical symbols are missing. The thread was written in MS Word using Mathtype for the symbols. I don't know what has gone wrong as I have used this method before without problems. I will try to correct the thread.

Matheinste.

 

[matheinste]

Correction to #5

----let us suppose a point to move in any way in a plane (that of x,y may be chosen). It describes some curve in this plane. If we draw this curve, we can, by looking at it, get an impression of the shape of its path, but not of any other data of its motion, e.g., the velocity which it has at different points of its path, or the time at which it passes through these points. But if we add time, t as a third coordinate, the same motion will be represented by a three dimensional curve, the form of which immediately gives us information about the character of the motion; for we can recognize from it which t belongs to any point x,y of the path, and we can also read off the velocity at any moment from the inclination to the x-y plane. We shall follow Minkowski by appropriately calling this curve the world-line of the point. A circular motion in the x-y plane would be represented by a helical world-line in the x-y-t manifold. This trajectory of the point only arbitrarily expresses, as it were, one aspect of its motion, viz. the projection of the tree dimensional world-line on the x-y plane. Now, if the motion of the point itself takes place in three dimensional space, we obtain for its world-line a curve in the four dimensional manifold x,y,z,t , and from this line all characteristics of the motion of the point can be studied with the greatest of ease. The path of the point in space is the projection of the world-line on the manifold x,,z,t and thus gives an arbitrary and one-sided view of a few properties only of the motion: whereas the world-line expresses them all in their entirety.----

Matheinste.

 

[robphy]

Here is the passage from Schlick (via Google books)
http://books.google.com/books?id=vj...+us+suppose+a+point&source=gbs_search_s&cad=0