Re: [Phys-l] Re. Simultaneity

[Moses Fayngold <moshfarlan@yahoo.com>]

--- On Thu, 5/21/09, Michael Edmiston <edmiston@bluffton.edu> wrote:
> "Students misinterpret the diagrams because they have been taught an 
> improper way to find the projections of points onto the graph's axes".

 I would not call it "improper", but rather a very special case. And I agree
with Michael that this should be explicitly stated from the onset.
  
 >   "Although this method works when the axes are orthogonal, this method does > not work when the axes are not orthogonal ..."
> The proper way to find the projection of a point onto the x-axis to draw a 
> line through the point such that the line is also parallel to the y-axis 
> (not perpendicular to the x-axis.).  The fact that the perpendicular method 
> works with the conventional graph is an artifact of the axes being 
> orthogonal.  The perpendicular method is not the general method for finding 
> the projection of a point onto a particular axis of that graph."

 So we agree 100% on this.

> "...when dealing with a two-dimensional plot for which the axes may or 
> may not be perpendicular, it is not the correct way to find the x and y 
> values.   Maybe I shouldn't be using the word "projection."  "

  The term "projection" is still used, but in the case described by Michael, it is called the oblique projection, as opposed to the perpendicular or "normal" projection. Both projections, while being different, are legitimate (and necessary)
characteristics of an event or a 4-vector between two events even in the case when the axes are not perpendicular (see below). 

 >  "But regardless of how we say it, if you try to identify the position and time >values of a point on a spacetime diagram by using perpendicular construction >rather than parallel construction, you will misinterpret the diagram."  

  While being essentially agree with Michael, I would still take a more conservative
stand. As I said, BOTH  kind of projection - normal and oblique - are important characteristics of an event or of the corresponding vector. If we want to find the moment of time of the event as measured by another observer whose coordinate axes are not mutually perpendicular when considered in our RF, we must use the oblique projection of the event onto the time-axis of the moving observer. Similarly, if we want to find the x-coordinate of this event as measured by the moving observer, we will correctly predict the outcome of his/her actual measurement by taking the oblique projection of the event onto his/her x-axis. In either case it is the OBLIQUE projection that represents the physically measured characteristics, and Michael is 100% right in this respect.
 But we should not forget the other side of the coin - that both described characteristics, while being directly measurable and thereby giving the most straightforward description of reality, are at the same time relative - different for different observers. On the other hand, there is less straightforward, more abstract characteristic but having the advantage of being absolute (invariant) - the interval associated with the event (or the norm of the corresponding 4-vector). In view of it being an invariant characteristic, it is in some respects even more important than just spatial or temporal coordinate of the event. We can still find this characteristic using only the oblique projections, but we can also find it using only normal projections. In the first case we say that the 4-vector is represented by its contravariant components, in the second case - by its covariant components. There is no discrimination against one or the other. In both cases the
 found norm is the same, but it is not positively defined - the squares of spatial and temporal coordinate enter the expression with the opposite signs (the pseudo-Euclidean metric). The symmetry and equal importance of both kind of projection is manifest in the existance of the third way to obtain the norm - the dot product of the vector with itself (or the dot product of any two vectors for that matter), in which we have the sum of products of co- and contravariant components of the vector. It gives again the same value of the norm but now all the products enter the expression with the same sign, and the symmetry between them is demonstrated in its full swing. 
  One can find a description of the co- and contravariant components of a vector in some books on General Relativity, but it is rather abstract there. A less abstract description can be found in the Course of Theoretical Physics (Vol. 2, the Classical Field Theory) by L. Landau and E. Lifshits. Another description (which I hope is even more simple) with specific examples can be found in my book "Special Relativity and How It Works" (Sec. 4.7, 4-vectors and tensors).

Moses Fayngold,
NJIT  

  


 

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