Re: Metric Tensor of Flat Space-Time

From: Ken S. Tucker (dynamics_at_vianet.on.ca)
Date: 09/06/04 

Date: 6 Sep 2004 11:13:29 -0700

dralexgreen@yahoo.co.uk (Alex Green) wrote in message news:<42c8441.0409060132.14500e5d@posting.google.com>...
> dynamics@vianet.on.ca (Ken S. Tucker) wrote in message news:<2202379a.0409022003.55d4176c@posting.google.com>...
> > dralexgreen@yahoo.co.uk (Alex Green) wrote in message news:<42c8441.0409020309.78c2e2ec@posting.google.com>...
> > > dynamics@vianet.on.ca (Ken S. Tucker) wrote in message news:<2202379a.0409010354.97c135f@posting.google.com>...
> > > > suzysewnshow@yahoo.com.au (suzysewnshow) wrote in message news:<e0a23188.0408312137.224287c5@posting.google.com>...
> > > > > dynamics@vianet.on.ca (Ken S. Tucker) wrote in message news:<2202379a.0408300329.d67f30f@posting.google.com>...
> > > > [...]
> [snip]
> > > > But I
> > > > doubt the time axis is necessary for the physical
> > > > foundation of the universe as it requires memory.
> > > >
> > >
> > > I proposed earlier that there were 3 ways of using Gauss' analysis to
> > > get to:
> > >
> > > ds^2 = - dt^2 + dx^2 etc..
> > >
> > > 1. Time could be imaginary.
> > > 2. g00 could be assigned the value -1 at the outset (ie: the metric is
> > > assumed)
> > > 3. According to Gauss' analysis of surfaces dT/dt could be imaginary.
> > >
> > > See: http://www.users.globalnet.co.uk/~lka/tensors.htm
> > >
> > > Suppose the third option is true so that the g00=-1 in the metric
> > > tensor for flat space-time originates in (idT/dt)^2 where dT is a
> > > small interval in the observer's coordinate system and dt is a small
> > > interval on the observed space-time surface.
> > >
> > > If it is assumed that the surface is flat and x,y,z,t,T are
> > > displacements from the origin then:
> > >
> > > s^2 = idT/dt idT/dt tt + xx + yy + zz
> > >
> > > s^2 = -t^2 + x^2 etc..
> > >
> [snip]
> > What you're discussing is like radar ranging.
> >
> > x^2+y^2+z^2+(ct)^2 is the radar distance for two objects
> > A and B at relative rest, separated by 'r'.
> > The diagonal line is the radars spacetime path originating
> > at A, then reflected back from B to A, received at 2t.
> >
> > A B at time=0
> > |\ |
> > | \ |
> > | \|
> > | /|
> > | / |
> > |/ |
> > A B at time = 2t, r=ct
> >
> > Seems that
> > s^2 = x^2+y^2+z^2+(ct)^2 = r^2 + (ct)^2
> > is the spacetime distance.
> >
> > This is compatible with ds^2 = g_uv dx^u dx^v
> > if one uses g00 =1, g11=1 and g01 = -v/c,
> > then one obtains,
> >
> > ds^2 = g00 (ct)^2 + 2g01 cdt dx + g11 dx^2.
> >
> > Subbing the metrics and (v = dx/dt) gives the familiar
> >
> > ds^2 = (ct)^2 - dx^2.
>
> Surely this only applies on the inbound path of the radar pulse where
> the velocity is -v and means that the pulse comes back to the radar.

"v" is velocity of A relative to B.
 
> >
> > Given a Euclidean space, the metrics are constant so
> >
> > s^2 = g_uv x^u x^v
> >
> > Again subbing the metrics,
> >
> > s^2 = g00 (ct)^2 + 2g01 ct x + g11 x^2
> >
> > = (ct)^2 - 2 (v/c) t x + x^2
> >
> > The relative velocity of A and B is v =0, so
> >
> > s^2 = (ct)^2 + x^2
> >
> > The radar pulse itself moves at the speed of light c.
> > In this case v=c = dx/dt =x/t, and this gives,
> >
> > s^2 = (ct)^2 - x^2 = 0,
> >
> > meaning the spacetime distance from the radar to
> > itself is zero, which is reasonable.
>
> Yes, agreed, but if 't' is real the space-time interval has a
> different topological signifcance from the concept of 'zero distance'.
> The 'causal light cone' summarises this topology, the surface of the
> backward cone being all those points where a light ray could have
> originated that could be at the origin at a particular moment.

Ok
 
> If 't' were based on units of (sqrt -1) then the causal light cone
> disappears and the backward cone becomes all those points that are
> also at the origin at a particular instant. Clearly this is physically
> incorrect, a photon cannot interact at two places at once.
>
> But what of the third option described above? Returning to the
> Gaussian derivation of metric tensors, suppose g00 is dT/dt where dT
> is the time interval on the tangent plane to a space-time surface and
> dt is the time interval on the surface itself. If the tangent plane
> has imaginary time intervals and the surface has real time intervals
> then g00 = -1 and
> the causal light cone reappears because all interactions take place on
> the space-time surface where time is real.
>
> See: http://www.users.globalnet.co.uk/~lka/tensors.htm

Did that.
 
> The third option would be indistinguishable from GR at the classical
> level but would have three advantages, firstly it would allow the
> existence of strange phenomena, such as wavefunctions, provided actual
> observed transfers of energy do not occur on the space-time surface.
> Secondly it would allow a set of things at (r,t) to also be at a point
> (0,0) in the observer's coordinate system (the tangent plane) provided
> no energy was transferred to the point. Thirdly time would exist in
> the observer as a geometric entity like space, this would allow the
> strange phenomena of conscious experience such as things in brain
> activity being present simultaneously in experience, the existence of
> an apparent observation point within the brain activity that is
> experience etc.
> Best Wishes
> Alex Green

I think most GRist's would agree with your reasoning, and
I have no problem with using it as an introduction and in
a weak field. It was the approach Einstein used in his 1914
introduction to the GR foundation. For brevity and clarity
of introduction he made some simplifying assumptions.
  
I've clearly presented an alternative spacetime,
using g_01 = -dx/dt, that is in accord with the ISU's
L=cT interdefinition of Length and Time using c,
without any reservations whatsoever, and therefore
fully support the ISU's decision and have no need
of any imaginary, physically ill-defined quantities,
nor do I recommend them in modern GR, not withstanding
the real possibilty that time is truly imaginary as
some think it should be on philosphical grounds and
so an imaginary time may yield deeper insight into our
physical laws. For me that argument is *fringe* now,
looked at it but the experimentalists clock sweeps
out real areas in real time, as do the planets.
Regards
Ken S. Tucker

Relevant Pages

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