Are dimensions also relative?

Answer

Dear Mehmet,

That is so, just like time, space also depends on the speed according to special relativity.

If you have two observers (numbers 1 & 2), with number 1 moving at a speed v along the x-axis relative to number 2, then in classical Newtonian mechanics time and place are connected for two observers as:

t2=t1

x2=x1+v*t

y2=y1

z2=z1

In special relativity there is 1 assumption that differs from Newtonian mechanics. Where Newtonian mechanics does not assume a maximum speed, this is the case with special relativity: namely the speed of light, this is the same for all observers. This ensures that time and space for our two observers are connected as follows:

t2=g(t1+v*x1/c²)

x2=g(x1+v*t)

y2=y1

z2=z1

with g=1/sqrt(1-v²/c²)

As you can see, both the x and t coordinates are adjusted by the speed via a scaling factor g (When you study these equations in detail, you can even see that the Lorentz transformation, as this set of equations is called, can actually be seen as a ‘kind of rotation’ converting space into time and vice versa).

The transformation given above has a few important consequences for observers moving at high speed relative to each other:

• Time Dilation: Time seems to move more slowly for moving observers.
• Length contraction : An object of length l moving at speed v is perceived with length l*g (Note, only if the movement is completely longitudinal)

The relativity of length is only in the direction of the movement, so a sphere will not shrink but will deform into a flattened sphere.

Time dilation and length contraction also give rise to a whole set of paradoxes, which give many a physics student headaches when working them out:

• twin paradox
• ladder paradox