Massive particles in de Sitter space

Can massive particles in de Sitter space move faster than light?

For the radial coordinate (in static coordinates) I have got the hyperbolic expression

$$r(\tau)\propto \sinh\left(\sqrt{\frac{\Lambda}{3}}\,\, c\, \tau\right)$$

with cosmological constant $$\Lambda$$, the speed of light $$c$$ and the proper-time $$\tau$$ of the particle. Accordingly, $$\dot r(\tau)$$ should become greater than $$c$$ after a certain time.

EDIT: The associated geodesic equation is

$$\ddot r-\frac{\Lambda c^2}{3} r=0$$

Metric:

$$ds^2=(1-k r^2)\, c^2 dt^2-\frac{dr^2}{1-k r^2}-r^2 d\Omega^2_2$$

with $$k=\Lambda/3$$.

• Can show the geodesic equation you got for radial component? Commented Feb 22, 2020 at 18:24
• Actually, it is not the most general one. I would appreciate to get one, which helps to solve my problem.
– user56224
Commented Feb 22, 2020 at 19:01
• $\dot{r}$ can be larger than c even in special relativity, this is not the local velocity but the celerity, which is the uncontracted coordinate length by contracted proper time. But in De Sitter space the velocity of a far away object can also be larger than c, particles behind the hubble radius can indeed recede faster than c, relative to an imaginary probe that is stationary to the central observer (behind the hubble radius only tachyons can fulfill the condition to be stationary to the central observer). Locally you can't exceed c though, see bit.ly/2SToXQx Commented Feb 22, 2020 at 19:30
• In the lecture he (Susskind) is telling us (somewhat handwaving) that "without dark energy nothing would really move out of our ability to see...." So, what if there is no dark energy...?!?
– user56224
Commented Feb 23, 2020 at 18:04
• Maybe we are talking about the same thing and the pass through the horizon is just this simple geodesic completion.
– user56224
Commented Feb 23, 2020 at 18:27

Firstly let's find a general geodesic equation (this part is not actually related to the question but you asked for it in the comments). Taking $$\phi=const$$ and $$\theta=\pi/2$$ the Lagrangian becomes $$L=(1-r^2/l^2)\dot{t}^2-\frac{1}{(1-r^2/l^2)}\dot{r}^2$$ Where $$l$$ is the de sitter horizon. Substituting this in the Euler Lagrange equation we get two equations, one for $$t$$ and one for $$r$$. Solving these equations one can write the radial equation as $$(l^2-r^2)\ddot{r}-ra^2+r\dot{r}=0$$ $$a$$ is a constant you'll get when you solve the time geodesic. Solving this and substituting appropriate initial conditions one gets $$r(\tau)=l\cosh{(\tau/l)}-\frac{la^2}{2}e^{\pm \tau/l}$$ Taking the derivative $$\dot{r}=\sinh{(\tau/l)}\mp\frac{a^2}{2}e^{\pm \tau/l}$$

Coming to your actual question now. It doesn't matter if the particle is traveling faster than the speed of light until you can observe it doing so. Long before the particle reaches the speed of light it will be beyond the de sitter horizon (you can calculate this using the formulas given above). The exception is when $$a=1$$. In that case it can be shown that the particle never reaches the speed of light.

Hope this helps.

• Your solution for the radial coordinate also satisfies the equation of my post. Moreover, the second part of your answer is correponding to the view I have seen in some textbooks and therefore certainly not unjustified. However, I still struggle with this sort of argumetation. I find it even hard to accept that the particle could escape the 3-sphere. I my opinion there must be a clearer explanation.
– user56224
Commented Feb 23, 2020 at 4:15
• @kaffeeauf Yes I know. I was wrong in saying that your solution is wrong that is why I deleted my comment. I had posted that comment because the differential equations are very different. Commented Feb 23, 2020 at 4:50
• @kaffeeauf Why do you think that it won't cross the horizon? Commented Feb 23, 2020 at 4:50
• The metric under consideration is (incompletely) correponding to the de Sitter space with closed spatial time-slices. These foliations are isometric to a 3-sphere. In global coordinates every point of this 3-sphere is reachable for a particle. Moreover, the 3-sphere has no boundary, so there is nothing left for a particle to escape from.
– user56224
Commented Feb 23, 2020 at 5:15
• @kaffeeauf The geodesic equations have been calculated for static coordinates which clearly have a coordinate singularity at $r=l$ which can be escaped. If you want to discuss what happens in global coordinates you must represent the geodesic equation in that coordinate and see what you get. In global coordinates the particle is always on the surface of a 3-sphere and a horizon makes no sense. Commented Feb 23, 2020 at 7:11