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- TL;DR Summary
- Validity of some descriptions shown in the Web where time is a vector in Galilean relativity

I refer to the video of this page, where there is a description of Galilean relativity that is meant to be an introduction to SR, making the comprehension of the latter easier as a smooth evolution from the former.

All the series is in my opinion excellent, but I think that this aspect is flawed.

It seems that the idea of the video is assuming that also under Galilean relativity there is spacetime as a vector space, composed of time and space vectors, which would be spanned by bases composed of a time basis vector

I will explain what the video does, with some adaptations, to make it simpler.

Imagine that the platform is a blue basis and the train is a red basis. The train is displacing wrt to the platform at v = ½ m/s. Event 1 is when the mid-points of the platform and train are instantaneously aligned, and we fix at that point the origins of the two reference frames. Event 2 happens at (common) time t = 2 s, but at x = 1 in the train frame.

To infer the Galilean transformation from red into blue, one can, for example, proceed as follows:

- Measure the (origin) red basis vectors in terms of the (destination) blue basis vectors (assumed to be unitary), as follows:

[tex]\color{red}{e_x} = (\color{blue}{e_x},0) = (1,0)[/tex]

[tex]\color{red}{e_t} = (\color{blue}v{e_x},{e_t}) = (v,1)[/tex]

- Put these values as column vectors of the change of basis matrix:

[tex]\left( {\begin{array}{*{20}{c}}

\color{blue}{x}\\

\color{blue}{t}

\end{array}} \right) = \left( {\begin{array}{*{20}{c}}

1&v\\

0&1

\end{array}} \right)\left( {\begin{array}{*{20}{c}}

\color{red}{x}\\

\color{red}{t}

\end{array}} \right) = \left( {\begin{array}{*{20}{c}}

{1*\color{red}{x} + v*\color{red}{t}}\\

{0*\color{red}{x} + 1*\color{red}{t}}

\end{array}} \right) = \left( {\begin{array}{*{20}{c}}

{\color{red}{x} + v\color{red}{t}}\\

\color{red}{t}

\end{array}} \right) = \left( {\begin{array}{*{20}{c}}

{\color{red}{1} + 0.5*\color{red}{2}}\\

\color{red}{2}

\end{array}} \right) = \left( {\begin{array}{*{20}{c}}

\color{blue}{2}\\

\color{blue}{2}

\end{array}} \right)[/tex]

The video also talks about an invariant spacetime interval which would be the arrow joining events 1 and 2, built as the addition of the time and space intervals in each frame, with this mathematical expression:

[tex]\vec S = \color{red}{x{\vec e_x}} + \color{red}{t{\vec e_t}} = \color{blue}{x{\vec e_x}} + \color{blue}{t{\vec e_t}}[/tex]

However, despite the beauty of the attempt, my opinion is that it obscures instead of clarifying the transit from Galilean to SR. The shift of paradigm represented by SR is precisely that in the Galilean context the vectors existed only for the spatial dimensions, time being a different thing, not a homogeneous unit that you could combine with space units.

Some thoughts to support this claim:

- The above-mentioned Galilean spacetime interval is useless, since it does not solve any practical problem.

- One can say that the “metrics” of space and time in Galilean relativity are different. But why? In SR we homogenize time and space units by multiplying time by the invariant speed of light c. There is no invariant speed in the Galilean framework. Anyhow, the author of the video claims in some comments that he can multiply space units by any arbitrary speed, preferably unitary to keep the numerical value of the time units, like c = 1 m/s. I am convinced that this is unacceptable but it would be interesting to specify why.

Which position would you take and how would you support it?

EDIT: I have later realized that in the next session of the series the author does admit that there is no sensible Galilean spacetime "distance". In fact, he talks about a ST "separation" vector. The question is still why to talk about a ST "whatever" vector containing components that are not recomposed to find any problem-solving magnitude.

All the series is in my opinion excellent, but I think that this aspect is flawed.

It seems that the idea of the video is assuming that also under Galilean relativity there is spacetime as a vector space, composed of time and space vectors, which would be spanned by bases composed of a time basis vector

**e**and another (simplifying the three spatial dimensions into one)_{t}**e**._{x}I will explain what the video does, with some adaptations, to make it simpler.

Imagine that the platform is a blue basis and the train is a red basis. The train is displacing wrt to the platform at v = ½ m/s. Event 1 is when the mid-points of the platform and train are instantaneously aligned, and we fix at that point the origins of the two reference frames. Event 2 happens at (common) time t = 2 s, but at x = 1 in the train frame.

To infer the Galilean transformation from red into blue, one can, for example, proceed as follows:

- Measure the (origin) red basis vectors in terms of the (destination) blue basis vectors (assumed to be unitary), as follows:

[tex]\color{red}{e_x} = (\color{blue}{e_x},0) = (1,0)[/tex]

[tex]\color{red}{e_t} = (\color{blue}v{e_x},{e_t}) = (v,1)[/tex]

- Put these values as column vectors of the change of basis matrix:

[tex]\left( {\begin{array}{*{20}{c}}

\color{blue}{x}\\

\color{blue}{t}

\end{array}} \right) = \left( {\begin{array}{*{20}{c}}

1&v\\

0&1

\end{array}} \right)\left( {\begin{array}{*{20}{c}}

\color{red}{x}\\

\color{red}{t}

\end{array}} \right) = \left( {\begin{array}{*{20}{c}}

{1*\color{red}{x} + v*\color{red}{t}}\\

{0*\color{red}{x} + 1*\color{red}{t}}

\end{array}} \right) = \left( {\begin{array}{*{20}{c}}

{\color{red}{x} + v\color{red}{t}}\\

\color{red}{t}

\end{array}} \right) = \left( {\begin{array}{*{20}{c}}

{\color{red}{1} + 0.5*\color{red}{2}}\\

\color{red}{2}

\end{array}} \right) = \left( {\begin{array}{*{20}{c}}

\color{blue}{2}\\

\color{blue}{2}

\end{array}} \right)[/tex]

The video also talks about an invariant spacetime interval which would be the arrow joining events 1 and 2, built as the addition of the time and space intervals in each frame, with this mathematical expression:

[tex]\vec S = \color{red}{x{\vec e_x}} + \color{red}{t{\vec e_t}} = \color{blue}{x{\vec e_x}} + \color{blue}{t{\vec e_t}}[/tex]

However, despite the beauty of the attempt, my opinion is that it obscures instead of clarifying the transit from Galilean to SR. The shift of paradigm represented by SR is precisely that in the Galilean context the vectors existed only for the spatial dimensions, time being a different thing, not a homogeneous unit that you could combine with space units.

Some thoughts to support this claim:

- The above-mentioned Galilean spacetime interval is useless, since it does not solve any practical problem.

- One can say that the “metrics” of space and time in Galilean relativity are different. But why? In SR we homogenize time and space units by multiplying time by the invariant speed of light c. There is no invariant speed in the Galilean framework. Anyhow, the author of the video claims in some comments that he can multiply space units by any arbitrary speed, preferably unitary to keep the numerical value of the time units, like c = 1 m/s. I am convinced that this is unacceptable but it would be interesting to specify why.

Which position would you take and how would you support it?

EDIT: I have later realized that in the next session of the series the author does admit that there is no sensible Galilean spacetime "distance". In fact, he talks about a ST "separation" vector. The question is still why to talk about a ST "whatever" vector containing components that are not recomposed to find any problem-solving magnitude.

Last edited: