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It was shown earlier by Rahaman et al. that a noncommutative-geometry background can account for galactic rotation curves without the need for dark matter. The smearing effect that characterizes noncommutative geometry is described by means of a Gaussian distribution intended to replace the Dirac delta function. The purpose of this paper is two-fold: 1) to account for the galactic rotation curves in a more transparent and intuitively more appealing way by replacing the Gaussian function by the simpler Lorentzian distribution proposed by Nozari and Mehdipour and 2) to show that the smearing effect is both a necessary and sufficient condition for meeting the stability criterion.

That noncommutative geometry can account for galactic rotation curves without the need for dark matter has already been shown in Ref. [

An important outcome of string theory is the realization that coordinates may become noncommuting operators on a

Here the mass

To connect the noncommutative geometry to dark matter and hence to galactic rotation curves, we need to introduce the metric for a static spherically symmetric spacetime:

For this metric, the Einstein field equations are

and

One goal of any modified gravitational theory is to explain the peculiar behavior of galactic rotation curves without postulating the existence of dark matter: test particles move with constant tangential velocity

where

To address the issue of stable orbits, we first note that given the four-velocity

which results in

Here the constants

From these conditions, we obtain [

The orbits are stable if

and unstable if

The smeared gravitational source in Equation (1) leads to a smeared mass. More precisely, the Schwarzschild solution of the Einstein field equations associated with the smeared source leads to the line element

The smeared mass is implicitly given by

which can also be obtained from Equation (3) (Equations (4) and (5) also yield

so that the modified Schwarzschild solution reduces to the ordinary Schwarzschild solution (see

The mass

Observe that the mass of the shell becomes

again dependent on

At this point we can finally address the question of stability by examining the potential

So from Equation (8),

To see the effect of the smearing, we first compute

From Equation (17),

and

It now follows directly that at

We therefore have a stable orbit at

We saw in the previous section that the smearing effect in noncommutative geometry is responsible for the stable orbit at

To this end, we return to Equation (20) and observe that the third term,

strongly dominates near

Hence

As noted above, for

To this end, we obtain from Equation (25),

and hence from Equations (21) and (22),

for every fixed

as a function of

So, as a next step, we plot

Recall that

It is shown in Ref. [

That noncommutative geometry, which has all the appearances of a small effect and can account for the galactic rotation curves, is consistent with the corresponding situation in