There are several alternatives to consider when trying to understand the cause of the Galactic rotation anomaly. One of these is of course, the application of CDM to maintain Newtonian behaviour. However, the invention of CDM ignores the advice of Occam's razor which states that for any two theories with correct outcomes, the one with the least number of new entities should be considered the correct theory. Implicit in this is the view that the theory which more closely matches known science is the most likely the correct one. CDM is currently the closest match with accepted science.

However, we may not have not pressed hard enough to identify other options which explain the anomaly and which conform to Newton, yet which do not require any new entities or any deviation from known science.

There is such an option that apparently has not been considered, which does not require the existence of CDM and which matches known science prefectly.

Postulate;-

We have underestimated the mass of the central supermassive black hole and as a result,

FIRSTLY:- all stars are rotating faster than Newtonian rules predict, due to the larger mass at the galactic centre.

SECONDLY;- we have not considered the effects of time dilation on Newtonian speeds when making observations from the Earth's frame of reference.

All gravitational fields exhibit time dilation which varies in degree, depending upon the elevation in the field, (distance from the mass). We can therefore also consider this effect to be a field. A spherical time rate field. The spherical time rate field around any mass is similar to a gravity well. It can be best envisaged by first imagining that we compress the body down to the density of a black hole. For the Earth, this radius is 4.5mm, the size of a grape. The curve or graph of the time rate against radius is always negative in value and is in the second quadrant. It rises steeply, close to and almost parallel to the vertical time axis, from zero at the black hole's radius, towards the origin. It then turns abruptly, proceeding almost horizontally and sloping up asymptotically to the "r" axis. It never quite reaches the "r" axis meaning there is always some residual time dilation all the way to infinity. The curve is basically the same shape as an inverse square curve but it is much more of a "sharper corner".

The equation of the curve is t=ROOT(1-2GM/rc^2) from the Schwarszchild metric. In 2D, imagine an inverted cymbal with a narrow downpipe in the middle. In 3D, visualisation, however difficult, becomes better than language. The 2D analogy is adequate.

Now, for relatively short radii from the centre of any mass and even for those at solar system scales, we do not notice much physical effect from the time dilation diminishing with "r". As an example, the Pioneer Anomaly is due to this effect and I have calculated (predicted) a blue shift of 9 x 10^-9 as opposed to the observed 6 x 10^-9, which is pretty close astronomically speaking. Whether or not this is the complete explanation for the anomaly, this effect demonstrates the arguments and helps us to visualise the time rate field and its effects.

When we look outwards from any position in the field, we observe blueshift and when we look inwards toward the middle of the field, we observe redshift. These shifts will affect our observation of any motion within the frame of reference where the motion is taking place. If we look outwards to a blue shifted region, speeds will appear faster than they actually are within the frame of the motion. If we look into a redshifted region speeds will appear slower.

When we observe galaxies, we are taking a view over much, much larger distances than solar system scales. We might then envisage that the relentless continuation of this time rate increase (time dilation decay), from the galactic centre outwards, will accumulate over these immense distances and so become significant in terms of the red and blue shift of Newtonian rotation speeds. This also explains why we have not yet detected the radial extent of all the galactic mass since this effect goes on to infinity regardless of where the mass ends.

- The only radial position that shows us a REAL, unshifted Newtonian rotation speed, has the same frame of reference as we have and so it has zero time dilation relative to us. This position is therefore at a radius similar to our own position in the Milky Way (for galaxies of similar mass and distribution).

- We must believe in Newton but we must also believe our observations. The Newton curve, therefore MUST pass through this point on the observed curve at a similar galactic radius as ours and which has the same time rate as we have.

So, we need to raise the calculated Newton curve so it crosses the observed curve at this position. We therefore deduce there is more mass at the centre, since all speeds are now greater, having raised the Newton curve.

In this raised position, the Newton curve now seems too high inboard and too low outboard of this "datum", compared with observation.

However,

- Inboard of this "Datum Radius", all Newtonian speeds are redshifted and slowed down relative to our frame of reference, increasingly so, as you look closer toward the galactic centre. The Newton curve inboard, therefore becomes increasingly lowered from the inverse square form as you move inwards and this brings the Newton curve down to match the observed.

- Outboard of the Datum Radius, all Newtonian speeds are blueshifted relative to our frame and so appear increasingly faster than Newton with increasing "r". The inverse square curve therefore becomes increasingly raised as you move outwards, bringing the Newton curve up to match the observed "flattened" curve.

- both these effects, inboard and outboard, result in a good "fit" between time shifted Newtonian speeds and the observed curve of rotation speeds.

If we imagine any particular frame of reference, either inboard or outboard from the Datum, we now see that the speed within that frame, any frame, always behaves in accordance with Newton, but it just does not appear to do so from our frame since the time rate is different between frames and the Newtonian speeds are distorted.

You can get a good picture of this if you go to "astronomy notes" and take the rotation curves from there. Print them, then cut out the expected curve and position it, crossing the observed curve at the solar sytstem radius. Draw this curve at this position over the observed curve. The effects outlined above now seem obvious, especially when you consider how long the horizontal "r" axis really is compared to the vertical axis of rotation speed. For matching scales on each axis, with equal kilometres, the differences between Newtonian speeds and adjacent, shifted Newtonian speeds are miniscule, but they become significant at galactic scales.

For this postulate,

We have not introduced any new entities.

We have not changed any known laws of science. (Newton and time dilation are both accepted phenomina).

We have simply used Newton and General Relativity to logically and conclusively deduce the effect. The arguments used are all mainstream, and the success or otherwise of this postulate depends simply on the numbers.

I have calculated the correct order of magnitude redshift and blueshift to match Newton with observations and so this postulate must now take precedence over the CDM proposal, in accordance with Occam's razor.

Pioneer Anomaly;-

We may as well consider the Pioneer Anomaly since this is due to exactly the same cause and effect.

Imagine the probes for the Pioneer anomaly, with their unexpected acceleration towards the Sun calculated from an observed blue shift of 6 x 10^-9 s/s.

From Earth, the probes appear at a position, way off course, due to the continuing blue shift. If we were to board a very fast space ship and take a quick trip to the probe, then the observed position of the probe as viewed from the ship will gradually shift from the position as observed from Earth, to the unshifted position as predicted by Newton. This will happen gradually over the course of the journey.

If we then reverse direction, but keep looking at the probe, then the opposite will happen on our return trip and the observed probe position will change from the Newtonian prediction back to the blue shifted position. We could keep doing this indefinitely and we will keep getting the same result.

The probes are indeed behaving in accordance with Newton, but they just don't look like it from the Earth's frame of reference, or from any position inboard in the time rate field.

So what is real? The only thing we can say is real, is whatever is the case WITHIN any frame we consider.

More generally;-

"Reality is only an appropriate notion for certain observations made from within the frame under consideration."

If we make observations from another frame, as we invariably do, they will always be distorted by the relative time rates and this is also why we observe the flat curve of galactic rotation speeds.