It is observed that the time required for riders to complete a race in

Yowamushi Pedal anime espisodes is considerably longer than the time necessary in our own experience. One hypothesis for this disparity is that the Yowamushi Pedal riders are in a different reference frame than our own, traveling at a high speed and thus resulting in time dilation. Another hypothesis is that the Yowamushi Pedal riders are in the vicinity of a large gravitational field (e.g. a black hole) which would also account for the observation. Below, we calculate the conditions necessary for each of the hypotheses to hold true.

##
Hypothesis 1: time dilation due to velocity difference between reference frames

The relative velocity between the two frames can be calculated from observed data as follows:

1) Bicycle racers in our local frame easily achieve an average speed of 45 km/hr over the final 1 km of a flat race course.

2) Races in the Yowamushi Pedal frame have been observed to take two full episodes, each episode lasting 20 min.

The time dilation equation is

$
\Delta t' = \frac{\Delta t}{\sqrt{1 - \frac{v^2}{c^2}}}
$

where $\Delta t'$ is an observed time period in the remote frame, $\Delta t$ is an observed time in the local frame, $v$ is the relative velocity between the two frames, and $c$ is the speed of light.

The equation can be re-arranged to solve for $v$:

$
v = c \sqrt{1-(\frac{\Delta t}{\Delta t'})^2}
$

From (1) above, the time $\Delta t$ in our local frame to cover 1km at 45 km/hr (=12.5 m/s) is 80 s.

From (2) above, the time $\Delta t'$ we observe cyclists to cover 1 km in the Yowamushi Pedal frame is 40 min = 2400 s.

We'll assume the two groups are riding at the same speed, and thus cover the distance in the same amount of time in their respective reference frames.

Thus, the relative velocity between the two frames that accounts for the time dilation is:

$
v = c \sqrt{1-(\frac{80}{2400})^2}
$

$v = 0.9994c$

So, the Yowamushi Pedal characters are *hauling ass* compared to us, traveling very near to the speed of light, even if it does take them nearly forever to cross that damn finish line!

## Hypothesis 2: time dilation due to gravity

Gravitational time dilation is equivalent to velocity time dilation in far space, using the gravitational escape velocity. Escape velocity $v$ from gravity field $g$ is

$v = \sqrt{2GM/r}$

Since we've already calculated $v$ above, we can compute the mass $M$ or the distance $r$ from the gravity field.

Let's assume a stellar-mass black hole, of 50 solar masses (this is about in the middle of the 10-100 solar mass range of stellar-mass black holes.) 1 solar mass = $1.989 × 10^{30} kg$, so

$M = 99.45 x 10^{30} kg$

The gravitational constant $G$ is

$G = 6.67408 × 10^{-11} m^3 kg^{-1} s^{-2}$

$c = 299,792,458 m/s$

From above,

$v = 0.9994c = 299,612,582 m/s$

Then,

$r = 2GM/v^2$

$r = 147,878 m$

Which is pretty darn close!

The Schwartzschild radius is the distance from the singularity to the event horizon (where $v_{escape} = c$):

$𝑅_𝑠=2𝐺𝑀/𝑐^2$

For our 50 stellar mass case,

$R_s = 147,701 m$

$R_s / r = 0.9988$

So, our Yowamushi Pedal riders could be falling in to a largish black hole, and would be right on the hairy edge of the event horizon. Note that the visual and physical distortions present close to a black hole, would also account for the warped appearance of the character Midosuji

## Conclusion

The observed time dilation could be accounted for by either of our hypotheses (e.g. due to velocity or gravity), or some interesting combination of the two. Midosuji's warped manifestation lends weight to the gravitational hypothesis, however.