# Scalar multiple of inertial frame

Consider an inertial frame of reference $$S$$. Now take a second frame $$S'$$, defined as follows: If a point $$P$$ has coordinates $$(t,x_1,x_2,x_3)$$ in $$S$$, then it has coordinates $$(t,2x_1,2x_2,2x_3)$$ in $$S'$$. Now my question is: is $$S'$$ an inertial frame of reference?

I get contradicting results by using the definitions from Wikipedia. On the one hand it says:

(...)it is a frame of reference in which Newton's first law of motion holds.

Certainly this law holds for $$S'$$, so it is an inertial frame. But then Wikipedia says:

Measurements of objects in motion (but not subject to forces) in one inertial frame can be converted to measurements in another by a simple transformation - the Galilean transformation in Newtonian physics or by using the Lorentz transformation in special relativity(...)

There is no Galilean transformation or Lorentz transformation that takes $$S$$ to $$S'$$. So $$S'$$ is not an intertial frame.

You are mixing frames and coordinate vectors, and that is leading to some confusion.

In a frame, vector quantities like position and velocity have some vector value. Changing frames may change those vectors (such as how accceleration is changed by transforming into a rotating frame).

When you speak of the coordinates changing from $$(t, x_1, x_2, x_3)$$ to $$(t, 2x_1, 2x_2, 2x_3)$$, you are using coordinate vectors. You construct a coordinate vector by starting with a frame and then defining a basis -- 3 vectors with which you are "measuring" all vectors. Any vector is thus converted to a coordinate vector by taking the dot products with all 3 vectors. The result is a triple (a quad, if you count time) of real numbers.

At the deepest level, your two "frames" are actually the same frame, with different trios of basis vectors. This means you may measure things differently, but fundamentally they are the same vector.

This is no more poignant than the fact that I can measure something as 1 inch long or 2.54 cm long. Fundamentally, the vector from one end of the object to the other didn't change. All I did was change my basis vectors, scaling by "units."

• Thank you, this is much enlightening. Another hint that these transformations are different, is that they do not preserve the spacetime interval. Commented Mar 3 at 11:17

Yes. This is still an inertial frame. The transform you have posted is a change of units for the spatial coordinates.

The Wikipedia list is not exhaustive.

• +1 for accuracy and lack of padding! Commented Mar 2 at 21:45
• Actually, translations and rotations are part of the Galilean transformation. Commented Mar 3 at 10:28