ROTATION CURVES. GEODESICS AND TANGENTIAL VELOCITY.
Rosemberg Toala Enriquez
In this note we will study the fall-off condition for the tangential velocity of an orbiting body.
It is known that the rotation curve for a galaxy is almost horizontal at large distances, (see
Figure 2.1), in contradiction with Newtonian gravity. This motivates the study of idealised systems where this behaviour can be explained in the context of General Relativity.
Figure 1: A collection of rotation curves from [3].
We are interested in stable and axisymmetric spacetimes since they represent a simple model for a rotating body such as a galaxy. Recall the following:
Definition.
A spacetime ( M, g ab
) is
• axisymmetric if there exists a spacelike vector field χ whose integral curves are closed.
• stationary if there exists a timelike Killing vector field ξ . Moreover, we say it is static if in addition ξ
⊥ is integrable.
Here we will always assume the spacetime ( M, g ab
) to be stationary and axisymmetric. Under these conditions we can choose a (non-unique) slicing Σ t
= { t = const } , where t : M →
R is a function such that dt ( ξ ) = 1, i.e., it is compatible with the flow of the stationary field.
Definition.
Let γ : I → M be a timelike curve, that is, a test particle.
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• We say that the particle is stationary if it sees an unchanging geometry as it moves, formally, if γ is parallel to a Killing field, i.e., ˙ = aξ + bχ , with a, b constants. Moreover, we say it is at rest with respect to Σ t if γ ˙ ⊥ Σ t
.
• The particle is free-falling if γ is a geodesic.
Now we proceed to compute the angular velocity for such particles. Suppose we can choose polar coordinates such that the metric takes the form ds
2
= − g tt dt
2
+ g rr dr
2
+ g
θθ dθ
2
+ g
φφ dφ
2
+ 2 g tφ dtdφ (1) and it approaches to the Minkowski metric as r → ∞ .
In this coordinates, being stationary is equivalent to following a trajectory with ( r, θ ) constant. And at rest implies a vanishing angular momentum, that is h ˙
φ i t h ∂ t
, ∂
φ i +
φ
˙ h ∂
φ
, ∂
φ i = 0. However, from this last equation we can deduce a non-trivial angular velocity as measured from infinity (i.e. in these asymptotically flat coordinates)
Ω = dφ dt g
= −
= tφ g
φφ
φ
˙ t ˙
Hence, the particle is locally nonrotating with respect to Σ t
, nevertheless it has a nonzero angular velocity as measured from the far away stars, this is interpreted as an inertial dragging effect produced by the rotating source. Now let us note that in the asymptotically flat coordinates we should expect g tφ
= O (1 /r ) and g
φφ
= O ( r
2
), thus Ω = O (1 /r
3
).
On the other hand, for a free-falling particle we can also define an angular velocity as measured from infinity. Recall that we have the following conserved quantities associated with the Killing vector fields, h ˙ t i = E, h ˙
φ i = J.
By solving the above equations we find
φ t
˙
˙
=
= g
φφ g tt
1
+ g 2 tφ
( J g tt
+ Eg tφ
)
1 g
φφ g tt
+ g 2 tφ
( J g tφ
− Eg
φφ
) so the angular velocity as measured from infinity is
Ω =
= dφ dt
J g tt
φ
˙
= t
˙
+ Eg tφ
J g tφ
− Eg
φφ
2

and we can see it has two contributions, the ‘classical’:
J g tt
J g tφ
− Eg
φφ
= O (1 /r 2 ), and the dragging effect produced by the rotating source:
Eg tφ = O (1 /r 3 ). However we must be careful
J g tφ
− Eg
φφ when taking r → ∞ since J is a constant of motion for the particle, that is, it depends on the geodesic and therefore on r . So this analysis is useful only when understanding a single orbiting body.
For a description of the fall-off condition of Ω we need to look at the geodesic equation, recall that this is given in coordinates by x i
+ Γ i jk x j x k
= 0 .
Proposition.
For a metric of the form (1) the angular velocity of a bounded geodesic evaluated at its maximum radius satisfies
−
Γ r tφ
Γ r
φφ
− s
Γ r tφ
Γ r
φφ
2
−
Γ r tt
Γ r
φφ
≤ Ω | r max
≤ −
Γ r tφ
Γ r
φφ
+ s
Γ r tφ
Γ r
φφ
2
−
Γ r tt
Γ r
φφ
Proof.
A bounded geodesic has a maximum radius r = r max at which ˙ r ≤ 0, thus the r -component of the geodesic equation reduces to
¨ + Γ r tt t ˙ 2
+ 2Γ r tφ t ˙ φ
˙
+ Γ r
φφ
φ
˙ 2
= 0 and moreover
Γ r tt
+ 2Γ r tφ
φ
˙ t ˙
+ Γ r
φφ
φ
˙
2 t ˙ 2
≥ 0
Now, as we will see, Γ r
φφ is negative, so
− 2Γ r tφ
− q
4(Γ r tφ
) 2 − 4Γ r tt
Γ r
φφ
Γ r
φφ
≤ Ω | r max
≤
− 2Γ r tφ
+ q
4(Γ r tφ
) 2 − 4Γ r tt
Γ r
φφ
Γ r
φφ and we obtain the desired bound.
It is interesting to write this bounds in terms of the metric. Recall that the Christoffel symbols are given by
Γ i jk
=
1
2 g is
( g sj,k
+ g ks,j
− g jk,s
) for the metric (1) we obtain
Γ
Γ
Γ r r r tt tφ
φφ
= −
1
2 g rr g tt,r
= −
1
2 g rr g tφ,r
= −
1
2 g rr g
φφ,r
.
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Corollary.
The tangential velocity of a bounded geodesic of the metric (1) satisfies
− r g tφ,r g
φφ,r
− s g tφ,r r g
φφ,r
2
− r 2 g tt,r g
φφ,r
≤ ( r Ω) | r max
≤ − r g tφ,r g
φφ,r
+ s g tφ,r r g
φφ,r
2
− r 2 g tt,r g
φφ,r
For example, in Schwarzschild spacetime we have
( r Ω) | r max
≤ r
M r and for equatorial geodesics in Kerr spacetime
( r Ω) | r max
≤ −
M a r 2
+ r
M a r 2
2
+
M r
Conclusions.
When starting this analysis I expected that the inertial dragging effect arising from a rotating mass would contribute significantly to explain the rotation curves of galaxies, however the conclusion is that, even when considering General Relativistic effects, we must have g tt
= O (log r ) in order to explain an asymptotically constant tangential velocity. This accounts to either an increase of mass (dark matter) or to a cosmological effect. The latter seems more pausible to me due to the homogeneity of the tangential velocity along all galaxies, roughly around 150 − 300 km/sec.
References.
[1] Begeman K. C.
H1 rotation curves of spiral galaxies.
Astron. Astrophys. 223, 47-60,
1989.
[2] Rourke C.
Mach’s principle and the dynamics of galaxies.
Available from: http://msp.warwick.ac.uk/~cpr/paradigm/galactic-dynamics-v2.pdf
[3] Sofue Y., Rubin V.
Rotation curves of galaxies.
Annu. Rev. Astron. Astrophys. 39,
137-174, 2001, arXiv:astro-ph/0010594v2.
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