## The Schwarzschild Solution Saturates to Constant Velocity at Far Distances, the Newtonian Limit is a Special Case Only

**Authors:** Franz Unterleitner

The focus of this work is on an integration constant which appears when the geodesic equations for the Schwarzschild solution are solved directly.
The constant has a dimension of speed-squared and is interpreted as vacuum energy.
When it is set to zero, the solution is Newtonian in the far limit, otherwise it causes orbital velocities to be constant in the long range, explaining flat rotation curves.
The literature is rich of examples where the four geodesic equations are solved, but the constant was not produced.
This is because the direct solution one of the geodesic equations was avoided and an `equivalent` equation was solved instead.
In this work the direct solution is achieved by using an integrating factor which results in the hitherto unknown integration constant.
The existence of fossile disks in the velocity profiles of galaxies is interpreted as a consequence of the vacuum energy decreasing in time.
Again, in short: The geodesic equations in their geometric form are second order differential equations.
They deliver additional integration constants when they are integrated to first order.
Not all of the constants survive, but one of them does and when it is not deliberately set to zero it saturates velocities at far distances.

**Comments:** 8 Pages.

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### Submission history

[v1] 2018-03-10 03:12:47

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