# A Dark matter and spacetime metric

#### Angelika10

I'm wondering if the galactic rotation curves could be attributed to a deviation of the metric of spacetime from the ideal Schwarzschild metric.

The Schwarzschild-metric is a well tested good approximation for the regions near the central mass - but at the outer space, far away from the central mass, we only assume that the curvature of spacetime approximates to zero (the curvature approximates to zero, the spacetime approximates to the Minkowski-Space). Could a deviating curvature of spacetime in the "far away field" lead to an accelaration in comparison to the (assumed) vanishing curvature?

It's like the difference whether we 1. assume the spacetime to be Minkowskian without matter - just add matter and curvature to the empty flat space or whether we 2. assume the spacetime itself is curved to vanish in the outer regions.

A "vanishing" spacetime (3 space dimensions approximating zero (not 1 = Minkowski, as in the standard model) and time accelarating to infinite) would build a curvature which would lead to an accelaration in comparison to the "Minkowskian outer space" which we assume in the standard model.

If one uses the rotation curves of the outer stars (constant velocity, independent of the distance r to the central mass) to determine the metric (standard form, spherically symmetric) the resulting metric is something like the square root of (r to the power of 3) in time and its inverse in space.

Have you seen something like that anywhere in the literature?

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#### Arman777

Gold Member

Staff Emeritus
This sounds an awfully lot like a personal theory, but the answer is "no". This would lead to a modification of Newton as a function of position, and it is known that this does not match the data. If you go down this path, it needs to be a function of acceleration.

#### Angelika10

Thank you, Arman777, for the article - that's exactly what I was looking for.

@Vanadium: Of course it is not independent of acceleration. And of course it leads to a modification of Newton. There's nothing wrong with a modification of Newton - General Relativity is a modification of Newton. One should only be careful to maintain Newton's law at the "solar system" point of the universe.

Farnes just tried quite successfully to explain the rotation curves by negative masses. Negative Masses or just "less space" lead to the same maths.

Let's begin with spherically symmetric, general metric and v(r) = const. in the outer regions of the galaxies...

Staff Emeritus
And of course it leads to a modification of Newton.
Please read my whole sentence, This would lead to a modification of Newton as a function of position, and it is known that this does not match the data.

#### Angelika10

I don't understand why you think it would lead to a modification of newton as a function of position. If the shape of the Schwarzschild-metric is different as we believe it is, like in Vossos (the link Arman posted), than the modification of Newton would be a function of accelaration.
In the potential in Vossos, there is a point, where the Potential crosses the x-axis. This point is not dependent on position.

Staff Emeritus
That paper is unpublished, so we don't discuss unpublished papers here. More to the point it is missing a good fraction of its equations.

Galaxies are non-relativistic. Therefore the only term that matters is the 00 term. That in turn means that two non-relativistic particles at the same point see the same curvature - or force - irrespective of their accelerations.

#### Angelika10

We can easily derive a metric by assuming a constant (independent of radius r) velocity in the outer regions of the galaxies: The outer stars gauge that metric in that point of view.
That metric can be combined with the Schwarzschild-metric of the near field to get a complete metric. It's just the "curvature-way" of MOND.

I really do not get why you complain about "function of position". Not r distinguishes, where the "new metric" crosses the x-axis - but value of the central mass.

#### Angelika10

it should read "the metric crosses y = 1" (not "the x-axis").

"Dark matter and spacetime metric"

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