The True Nature of Space and Time

[LaurieAG]

Stephen Hawking and Roger Penrose had a series of lectures/debates published by Princeton Science Library (The Isaac Newton Institute Series of Lectures) as 'The Nature of Space and Time', ISBN 0-691-05084-8, in 1996.

 

On page 108, chapter 6 'The Twistor View of Spacetime', paragraph 2, Roger Penrose writes

Before doing this in detail, let us consider two important roles of the Riemann sphere in physics.

1.

...

2. Imagine an observer situated at a point in spacetime, out in space looking at the stars. Suppose she plots the angular positions of these stars on a sphere. Now, if a second observer were to pass through the same point at the same time, but with a velocity relative to the first observer then, owing to abberation effects, he would map the stars in different positions on the sphere. What is remarkable is that the different positions on the points on the spheres are related by a special transformation called a Mobius transformation. Such transformations form precisely the group that preserves the complex structure of the Riemann sphere. Thus, the space of lightrays through a spacetime point is, in a natural sense, a Riemann sphere.

...

 

The problem

The main question that I ask is this, If you consider light rays passing through the spacetime point without the stationary observer or the moving observer, there is only ever one set of real light 'paths' through that discrete point at any one discrete time.

 

Looking closer

If you reduce this discrete point to the infinitessimal level you will find that all the light rays that will be captured by any observer right on the discrete point, stationary or moving, will be in exactly the same position i.e. delta x away from the discrete observation point in the same angular position, at the time it takes a photon to travel delta x away.

 

Looking closer still

An observer with a velocity of c would reach the discrete point in the time it takes a photon to travel delta x, i.e. it would be delta x away when the photons are delta x away. If the velocity is half c it will be (delta x)/2 away from the discrete point while the incoming photons are still delta x away etc. If the observer is stationary the distance away from the discrete observation point will equal zero at all times.

 

The real problem

Because delta x is the limiting factor i.e. delta x is infinitessimal so nothing can be smaller. V/c * delta x = observer distance away when photon is delta x away. If the distance away from the discrete observation point is smaller than delta x it cannot be calculated with calculus based infinitessimal mathematics because this combination only works when V = c or V =0.

 

It seems quite obvious that there's some sort of a category error here.

 

Please keep any discussion to c, V, delta x and distance to the discrete observation point.

[LaurieAG]

The attached image is an unfocussed version of my avatar with a mirror (i.e. a feedback loop screen capture based on 3 separate angles of distortion).

 

Since V=0 is the only discrete (that we can experiment with) viewpoint that gives correct results to this problem maybe we should look at some observations from a truely stationary observation point, without rotational, solar or galactic spin, then maybe we would see a clearer picture of our universe.

 

At least a stopped watch is correct twice a day.

[freeztar]

How can something be delta x away? :hihi:

 

Delta x usually means change in distance/length. So how can it be used as a set measurement?

[LaurieAG]

Hi Freeztar,

 

Delta x usually means change in distance/length. So how can it be used as a set measurement?

 

In basic calculus proofs from first principles delta x is the (minimum) width of the sections that a curve (plot of a function f(x)) is divided up into so that the area under the curve can be calculated within the limits given. Velocity, acceleration and distance travelled are a good example of this core integral relationship.

 

As such anything smaller than delta x is really outside the scope of the inherrent structure of calculus and anything deriving itself from the same core is in the same boat. In the example I gave delta x is a distance because it is how far away a photon is from an observation point, just before it hits the observation point.

 

If you use the argument that the distance of the moving observer away from the observation point should be used for delta x then you are using a totally different calculation (non Newtonian calculus) to that used by the stationary observer. The appearance of 'change' probably comes from the mathematical process when limits are applied to a function, as the function (the plot of its curve) approaches its limits delta x stays the same regardless, as delta x is an infinitessimal amount.

 

How you got there is irrelevant to any observations made from that discrete point at that discrete time, even delta x away, unless you are an incoming photon.

 

BTW, have you read the book? Some interesting stuff on the Riemann sphere.

[LaurieAG]

The screens captured from this setup are the same as those seen by anybody in the room. They are a true Poincare section of the loop.

[LaurieAG]

All the screen captures are from the same basic setup along with one or two mirrors.

[LaurieAG]

It's interesting what you can do with an optical loop.