Re: Metric Tensor of Flat Space-Time

From: Alex Green (dralexgreen_at_yahoo.co.uk)
Date: 09/08/04 

Date: 8 Sep 2004 15:13:21 -0700

dynamics@vianet.on.ca (Ken S. Tucker) wrote in message news:<2202379a.0409080242.5924d87f@posting.google.com>...
> dralexgreen@yahoo.co.uk (Alex Green) wrote in message news:<42c8441.0409071436.1f51ead5@posting.google.com>...
> > dynamics@vianet.on.ca (Ken S. Tucker) wrote in message news:<2202379a.0409070415.8bd911b@posting.google.com>...
>
> > > The use of the sqrt(-1) in QM is a calculation
> > > instrument, just as the "j" operator (j=sqrt(-1))
> > > is used for convenience in calculating complex
> > > impedance in electronic circuits, it is helpful
> > > to articulate right-angles in phasor diagrams.
> > > Would a g_00 =-1 be applicable there?
> >
> > My take on sqrt -1 is that it can be used to model some aspects of
> > physical systems (cf: j). Maths using sqrt -1 is a descriptive tool in
> > science that cannot be replaced by real numbers in some circumstances
> > (such as qm amplitudes). Another role for sqrt -1 is 'geometrical' in
> > that it allows us to insert another direction for arranging things
> > that has an interesting relationship to directions based on real
> > numbers. Clearly we cannot measure phenomena that are based on
> > imaginary numbers where these cannot be resolved into real numbers.
> > Does this mean that phenomena such as quantum amplitudes prior to
> > 'collapse' do not exist or does it just mean we cannot observe them
> > directly?
>
> Again I think we're using a mathematical instrument
> to do a calculation then trying to see if it's physically
> real, like asking, does 1+1=2 exist prior to our calculator
> saying so. Well no, it doesn't exist until we have a
> reading of 2 on the calculator.
>
> > Do you know of any mathematical recipe where complex numbers can
> > always be reduced to reals?
>
> I think you mean complex units on x-axis orthogonal to t, for example,
> then s^2 = t^2 - x^2. The same result occurs when all the units are
> real, when t is nonorthogonal wrt x, t^2 = s^2 + x^2, so yes always
> a recipe when two axes are involved.

Before I discuss this I would like to point out that I am not
proposing that time is imaginary as such but that the differential
coefficient that appears in the metric tensor (for g00) is imaginary.

I agree with what you are saying numerically but topologically there
is a considerable difference between
0 = x^2 + (it)^2 and
0 = x^2 - t^2

The first expression denotes that there is literally no separation
between (0,0) and (x,t) along the vector that points in this
direction. The second expression denotes a movement from (x,t) to
(0,0). In the physical world that we measure the second equation is
evidently a correct description (although the proper time on a fast
moving object would tend to zero along the path x->0).

Returning to the tangent plane applied to the space-time surface of
the world, this plane is not measured, it is assumed. Furthermore it
is a plane in a hypothetical coordinate system belonging to the
observer. If time were imaginary on this plane we would have no way of
determining this except by predictions such as g00 = -1 if g00 =
idT/dt (where idT is a small time interval on the tangent plane and dt
is a small interval on the space-time surface).

However, my reasoning would be wrong if it were possible to construct
a tangent as a straight line grazing the space-time surface without
assuming that 'straight line' was a property of the observer's
coordinate system.

Best Wishes

Alex Green

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