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I - The Warp Drive metric:

The Warp Drive is a geometry in a spacetime $(\mathcal{M},g)$ given (in geometrized coordinates $c=G=1$) by the following metric tensor:

$$ ds^{2} = -dt^{2}+ (dx-v_{s}f(r_{s})dt)^{2}+dx^{2}+dy^{2}+dz^{2} \tag{1}$$

The basic interpretation is: suppose that we have a spaceship who can generate a spacetime curvature like $(1)$ around itself. This geometry generates a region called a "bubble" which have the property that the curvature vanishes outside the bubble and inside too (where the ship lies), resulting a "boundary of curvature ". Also, this structure, expand the spacetime in the backwards of the ship and contracts in front of it, this mechanism then allows the ship to travel around. So, the $v_{s}$ is the velocity of the ship seeing by a outside observer (an astronomer) and $f(r_{s})$ is a function that forces the behaviour of "curvature vanishes outside and inside" (which means that the spacetime is minkowskian outside the bubble and is minkowskian inside, but have a curvature the edges) .

II - Using the Warp Drive:

Now, in the reference $[1]$ we encounter the following gendaken situation:

Suppose two planets $A$ and $B$, quite separated $D$ from each other. Then, the spaceship wants to travel from $A$ to $B$ using the warp drive propulsion. Starting from surface of planet $A$, the ship uses fuel based engines to travel at constant velocity $V$ a distance $d<<D$ to get inside a orbital station for spool the warp drives properly. Then the ship arrives at the station and turn off the fuel based engines. After that, in the station, the captain lights the warp drive throttle and generates the spacetime geometry $(1)$.

The ship then moves under a uniform accelerated motion $a$ a $L$ distance, to another space station in the orbit of planet $B$ distant $d$ from the surface; the ship ports at the station and after some time the crew back to $A$ unsing the same mechanism (therefore it'a round trip scenario).

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III - Some Facts:

The travel time of a round trip from a star $A$ to $B$ is given by (accordingly observers in Minkowski spacetime)$[1]$:

$$ T = 2\Bigg(\frac{d}{V} + \sqrt{\frac{D-2d}{a}} \Bigg) \tag{2} $$

But from the point of view of someone inside the "bubble", the travel time is then given by$[1]$:

$$ \tau= 2\Bigg(\frac{d}{\gamma V} + \sqrt{\frac{D-2d}{a}} \Bigg) \tag{2}$$

Now, external observers are living in Minkowski spacetime given by the metric:

$$ds^{2} = -dt^{2}+dx^{2}+dy^{2}+dz^{2} \tag{3}$$

So, the proper time is related with coordinate time as:

$$\tau_{1} = \frac{T_{1}}{\gamma} \tag{4}$$

But, the ship inside the bubble, generates a spacetime geometry around the ship (which is seen by external observers) given by the metric $(1)$

And the proper time of this geometry (or the path that warp bubble perform between the stars -seen by external observers-) is then $[1]$:

$$ \tau_{2} = T_{2} \tag{5} $$

$IV$ - My Analysis of why $\gamma$ appears only on $\frac{d}{V}:$

Suppose then that someone on station of planet $A$ measured the travel time of the ship. So the total coordinate time $T$, which an astronomer measured, is then

$$T = T_{1} + T_{2} = 2\Bigg(\frac{d}{V} + 2\sqrt{\frac{D-2d}{a}} \Bigg)$$

Where $T_{1} = \frac{d}{V}$ is the time which corresponds the travel using "fuel engines" and $T_{2} = 2\sqrt{\frac{D-2d}{a}}$ is the time using the "warp drive". So the $T_{1}$ are measured using a travel in minkowski spacetime and $T_{2}$ are measured using a warp drive metric (because the astronomer on earth sees the bubble crossing the path $L$)

Now, to see which time the clocks of the ship are marking and considering that $T_{1}$ is a motion in minkowski spacetime, the astronomer calculate:

$$\tau_{1} = \frac{d}{\gamma V}$$

But now, the proper time $\tau_{2}$ corresponds a motion using warp drive metric and then $T_{2} = \tau_{2}$. Therefore:

$$\tau = \tau_{1}+\tau_{2} = 2\frac{d}{\gamma V} + 2\sqrt{\frac{D-2d}{a}} $$

$V$ - Question:

My question is simple: My arguments in the part $IV$ can explain why the $\gamma$ appears only on $\frac{d}{V}$?

$$* * *$$

$[1]$ https://arxiv.org/abs/gr-qc/0009013

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