In the definition of a « speed » by a coordinate ratio, v=x/t for example, relative between Galilean frames of reference, the velocity of light is an unattainable upper limit (for material objects).
But is this physical?
Relativity uses the concept of 4-velocity where the dynamic parameter is the proper time that we will denote T.
Let’s stay in RR, where we can integrate the « dt » in « t », « dx » in « x », « dS » in « S » and « dT » in « T » (it is linear) and consider only 2 parallel inertial frames of reference of relative velocity v = x/t
This gives: t= x/v.
The spacetime interval S, we have:
S² = c²T² = c²t²-x²
This gives c²T²= c²x²(1/v²-1/c² ),
With y = cste and z = cste, the 4-vector Uµ={dt/dT, dx/dT, 0, 0}
For the spatial component let us set : V = dx/dT = X/T
By setting the 4-speed: V = x/T
We obtain:
V² = v² /(1-v²/c²)
Which can be > c, if v is very close to c.
Let’s take the example v = 0.99 c we obtain:
V= 7c, approximately.
Since the proper time is the parameter that physically measures the time of the observer who follows this geodesic having a speed close to that of light, in the reference frame of the inertial observer of reference that we can, arbitrarily consider at « rest », we see that this « velocity » of 7c has a physical character.
In one year of his own time, which also measures his aging, he travels a distance of 7 light years in the « rest » frame of reference.
This, illustrated by the « paradox » of the Langevin traveler (twins), is its mathematical foundation.