Consider a four-dimensional spacetime with coordinates ##t x y z##. For any massive particle, the interval is given by
$$ ds^2 = -c^2\,dt^2 + dx^2 + dy^2 + dz^2. $$
To move faster than light in ordinary special relativity, we would require a timelike worldline to become spacelike, meaning the interval would switch sign from negative to positive for the same particle. A hypothetical way to achieve this is to propose a continuous deformation of the metric itself: define a smooth function
$$ \Lambda(\tau) = 1 + \alpha(\tau), $$
where ##\tau## is the proper time and ##\alpha(\tau)## is a carefully constructed function such that it remains infinitesimal for almost all ##\tau## but integrates to a nontrivial topological shift over a closed loop in spacetime. Insert this ##\Lambda(\tau)## into the metric components so that each time the particle completes a topological cycle, its effective velocity limit locally looks like
$$ c_{\text{local}}(\tau) = c\sqrt{\Lambda(\tau)}. $$
If we engineer ##\alpha(\tau)## to return to zero after a finite interval of ##\tau##, the geometry reverts to the original Minkowski form yet encodes a net displacement in the spacelike direction that exceeds ##c\Delta t##. In other words, upon completing this “loop,” the particle has advanced farther in space than light could in the same coordinate time.
To see why this might fool us into thinking we can exceed the speed of light, notice that for each small segment of proper time ##\Delta \tau##:
$$ ds^2 \approx -c^2 \bigl(1+\alpha(\tau)\bigr)\,d\tau^2 + \dots $$
so that the local speed limit depends on ##\alpha(\tau)##. Summed over a whole cycle, the path length in space can accumulate faster than ##c\Delta t##. In particular, the four-velocity still never violates
$$ \frac{dx^\mu}{d\tau}\,\frac{dx_\mu}{d\tau} = -c^2 $$
locally, but the global topology of the altered metric allows us to come back to the same ##\tau## while having covered more spatial distance than a simple Minkowski observer would measure as permitted in a normal inertial frame.
Mathematically, because
$$ \int_{\text{loop}} \nabla_\mu \alpha(\tau)\,dx^\mu \neq 0, $$
there is a net change in the local light cone tilt for the traveling particle, which effectively accumulates a superluminal displacement. However, this rests on the fiction of a nontrivial spacetime topology or metric deformation that we do not observe in flat Minkowski spacetime. The moment we impose standard special relativity’s geometry again (with ##\alpha(\tau) = 0## everywhere), the argument collapses, and we see there is no global inertial frame in which the particle’s velocity has exceeded ##c##.
So the original assumption of interpreting ##c^2## as a speed is flawed: ##c^2## in ##E = mc^2## is a conversion factor from mass units to energy units, not a physical velocity limit. The actual limitation is due to the invariant interval ##ds^2## and the requirement that massive particles follow timelike trajectories with negative intervals:
$$ ds^2 = -c^2\,d\tau^2 < 0. $$
Thus the “speed” ##c^2## being about ##34{,}500{,}000{,}000## miles per second is never a meaningful velocity for travel, but rather an artifact of squaring ##c##. This is why physically traveling faster than ##c## is still prohibited once the proper relativistic geometry is restored.
