# The Two-dimensional Doppler Effect

## The separation of 2-d facts from 4-d inferences.

The physical sciences are unnecessarily difficult and are likely to remain so until it is widely known that they could be simplified at a stroke by splitting into two parts, a 4 dimensional part for subjective theories of what can not be seen and a much simpler 2 dimensional part for objective facts seen on the 2-d screens in our eyeballs.

Here is a 2-d objective fact: The direction you look is also the direction of your past. Each metre you look takes you 3.3 nanoseconds (ns) further into the past. Stand between a pair of mirrors 1 metre apart to see a long column of your reflections at intervals of 1 metre and 3.3 ns. This 2-d fact was discovered in 1676 but lost while being found by being interpreted subjectively with 4-dimensions as the discovery that the unseen 4-d speed of light is 1/3.3 metres per ns. The 2-d fact is still lost and its 4-d subjective interpretation is treated as a fact. Facts are sacred and should be separated from subjective inferences as facts and comment are separated in good journalism.

## 2-d Science

This article is about the 2-d relativistic Doppler Effect. In 2-d science there is only one direction – the direction you look. In this direction distance and time are unified because every metre you look in this direction takes your sight 3.3 nanoseconds further into your past. If 3.3 nanoseconds is defined to be a metre of past then every metre of distance is accompanied by a metre of past and the constant ‘c’ which is their ratio, is 1.

There is no speed in 2-d science but only what two observers see when they look at each other when they are approaching each other and when they are withdrawing from each other. In the first case each sees the other’s clock to be running faster than his own clock and in the second case each sees the other’s clock to be running slower than his clock. If each defines the rate of his own clock as 1, he sees the rate of the other’s clock to be 1+S in the first case and 1-S in the second case.

That S is the sine of an angle can be deduced from the measurements made by traffic control cameras along roads. They take two photographs of a vehicle travelling along a ruler painted on the road and time the interval between the photographs with a clock. When the distance between the locations of the images of the vehicle and the interval between the photographs are expressed in metres, their ratio is S=sin(a).

We are thus led to fig 1, a symmetrical zig-zag line og gradients 1 and -1 inside a right-angled triangle whose smallest angle is S. That leads fairly directly to equations  and  which describe the 2-d Doppler Effect.

## The 2-d Doppler Effect

The diagram in fig 1 generates the two simple equations,  and  which describe the Doppler Effect.

Equations - in fig 1 define the parameters S, C and D of the Effect, while  and  describe the Doppler Effect. Fig 2: How to obtain  and  from  in fig 1. Fig 3: Three sets of C’s and D’s calculated from three values of S by using D2=(1+S)/(1-S) and C2=(1-S2) Fig 4: Red and blue shifts plotted against S. A red shift of -1 explains why black holes are black

Besides exhibiting a blue Doppler shift, an approaching object is seen to be getting bigger.  Besides exhibiting a red shift a a retreating object is seen to get smaller.

## Einstein’s theory of relativity

In 1905, Einstein discovered the key equations of the Doppler Effect including  and  but he used them to construct a 4-d theory of relative speeds, v, and the speed of light, c. His theory is close to the description of the 3-d Doppler Effect initially, but it soon diverges from it when S, C and D are replaced by Einstein’s symbols. Fig 5 shows how he changed the symbols of fig 4 in the early stages of his theory and fig 6 shows how he added new equations later. Fig 5: In the first row are the symbols Einstein used instead of those in fig 3. In the second row are the equations which replaced CD=(1+S) and C/D=(1-S). Fig 6: How Einstein made four equations from the two in fig 5.

The much greater complexity of Einstein’s theory than the RDE as observed in the pictorial record is illustrated in fig 7. Fig 7: How the 2-d Doppler Effect and Einstein’s 4-d Theory of Relativity compare for difficulty and complexity.

The elements of the 2-d Doppler limit are limited and provide no opportunity for improvisation whereas there are no limits on the ideas that Einstein fed into his theory, making it unfalsifiable.

Thank you for joining me

Stan Clough

Professor Emeritus of Physics, School of Physics and Astronomy, University of Nottingham, UK.