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To demonstrate Lorentz transformations mathematically, we assume that time is a dimension (via linear transformations, etc.), what physical argument requires us to do this?

Details for the reason for this question: we suppose two observers O and O' at a point M, and a lamp which turns on and off every second for a short period of time which is very very small compared to the second and a distance L of the two observers, if O' begins to move away from the source at the moment it receives a signal, a simple calculation gives

$$ct'=L+vt'$$ $$ct'=L'=\frac{L}{1-\beta}\;\;,\;\;\beta=v/c$$

i.e. $$L=ct'-vt'=k_{1}(L-vt)\;\;,\;\; k_{1}=\frac{1}{1-\beta}$$ if the observer O ' approaches the source, we find $$L=ct''+vt''=k_{2}(L+vt)\;\;,\;\; k_{2}=\frac{1}{1+\beta}$$ if the observer O ' moves away orthogonally to the radius vector of the light wave, we find ((Pythagoras) $$L^{2}=\gamma^{2}L^{2}(1-\beta^{2})=\gamma(L-\beta L).\gamma (L+\beta L)$$ $$L^{2}=\gamma(L-vt).\gamma (L+vt)=L_{1}L_{2}$$

with $$\begin{cases} L_{1}=\gamma(L-vt) \\ L_{2}=\gamma (L+vt) \end{cases}$$ they have the same form as the Lorentz transformations but have nothing to do with the observer O' !

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  • $\begingroup$ Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Physics Meta, or in Physics Chat. Comments continuing discussion may be removed. $\endgroup$
    – ACuriousMind ♦
    Commented Aug 16 at 14:50

3 Answers 3

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What physical argument to say that time is a dimension ?

The standard definition of "dimension" is the smallest number of attributes or co-ordinates that we need to uniquely identify a specific location or point within a space of similar locations or points.

We know that three spatial co-ordinates are not sufficient to uniquely identify an event. We know this because different events can have have the same spatial co-ordinates i.e. different events can occur at the same location in space. We need a fourth co-ordinate (at least) to uniquely identify an event, and we call this fourth co-ordinate or dimension "time".

In short, if time were not a dimension then we could not distinguish yesterday from today.

However, from the context of your question, I suspect you may be using your own non-standard definition of "dimension".

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  • $\begingroup$ $a x+by=0$ linear independence ==> a=b=0 , but we write $x\pm ct=0$, i.e. $a=1,b=\pm c $ ??? $\endgroup$
    – The Tiler
    Commented Aug 13 at 11:51
  • $\begingroup$ Geometry lessons, linear algebra and differential geometry, M.Postnikov. A Course Of Higher Mathematics Vol 3 1 Linear Algebra by V. I. Smirnov,...... $\endgroup$
    – The Tiler
    Commented Aug 13 at 12:10
  • $\begingroup$ And I'm not the only one, after I published my article (during covid19, I formalized an idea that's been bugging me for almost 30 years), I found another who has the same idea but he kept the mathematical structure ... ,hal.science/hal-02068970/document $\endgroup$
    – The Tiler
    Commented Aug 13 at 12:46
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    $\begingroup$ @TheTiler On a two-dimensional plane the only values of $a$ and $b$ that make $ax+by=0$ for all values of $x$ and $y$ are $a=b=0$, hence $x$ and $y$ are linearly independent. But if you restrict a point to lie on a line e.g. $x+2y=0$ then you lose one degree of freedom, the line only has one dimension, and $x$ and $y$ are no longer linearly independent because if you know $x$ you can find $y$ and vice versa. Similarly in relativity the world line of an object is one dimensional because we only need one co-ordinate to uniquely define the position of the object along its world line. $\endgroup$
    – gandalf61
    Commented Aug 13 at 12:55
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    $\begingroup$ @TheTiler I usually refrain from saying this to people who are physics enthusiasts/amateurs, but please stop this nonsense. You are manifestly a crackpot, and it seems to me you won't ever change. Your "inconceivable dose of mathematics" was certainly a piece of cake for any first-year college student. I'm sorry to be rude, but I know what it feels like not to be understood in my work, and I know that if you don't change your mindset, you won't go anywhere. $\endgroup$
    – Jeanbaptiste Roux
    Commented Aug 13 at 15:36
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You can turn the argument on its head. We know experimentally that time dilation and length contraction etc. are physical effects, and spacetime vectors and tensors are an elegant way of describing them.

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  • $\begingroup$ I constructed a theory which gives the same results from the same postulates but without linear transformations..., there are undecidable results in the theory of special relativity....:vixra.org/pdf/2006.0280v6.pdf $\endgroup$
    – The Tiler
    Commented Aug 13 at 9:18
  • $\begingroup$ The mathematical beauty of a theory is not a physical argument, there are mathematically true, physically false theories. $\endgroup$
    – The Tiler
    Commented Aug 13 at 10:05
  • $\begingroup$ @TheTiler Asking if we can construct spacetime tensors is not a physical question. It is a practical one. The only question is whether they are a useful tool, and indeed they are. There may be alternative mathematical constructions to describe the same physics. $\endgroup$
    – Rd Basha
    Commented Aug 13 at 13:45
  • $\begingroup$ My question has nothing to do with tensors, I know that maths is a factory for making molds into which ideas are poured, why the 4D mold? $\endgroup$
    – The Tiler
    Commented Aug 13 at 14:12
  • $\begingroup$ @TheTiler Vectors are just a subset of tensors. The 4D vector piscure is very useful. If you know of a more useful structure, then good luck trying to convince people. $\endgroup$
    – Rd Basha
    Commented Aug 13 at 14:31
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Before SR, I think that the word "dimension" was reserved for spatial dimensions. An event now in my place and another somewhere in Mars, $2$ minutes from now, has the same distance separation than if the event happens there $1$ minutes from now.

After SR, I can postulate reference frames for each case, with relative velocities such that both events are simultaneous. Each of these frames have a define spacial distance between the places of the events.

That is the meaning to use a 4-D distance, or space-time distance between events: depending on the coordinate time interval for 2 points with a given coordinate spacial distance, we get a define distance for simultaneous events.

If we add the time-like separation, there is always a define number associated with a coordinate distance and a coordinate time interval. That suggests to use the concept of metric, due to mathematics similarity of being able to know the conventional distance between points, by knowing its spatial coordinates.

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    $\begingroup$ Spacetime is not a physical entity, though. Stuff 5 meters to the left and five meters to the right exists, right now, but stuff that was 5 seconds in the past or that will be five seconds in the future does not. More generally, coordinates are not real. They are our imaginary maps to the world. Nature doesn't know about them and nature doesn't care about them. The only physics that nature cares about are irreversible energy exchanges. That is what a clock produces: series of irreversible energy exchanges and those are what we call "time". $\endgroup$
    – FlatterMann
    Commented Aug 14 at 1:41
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    $\begingroup$ @FlatterMann thats not really true.Everything is relative so when you say 5 meters to the right ir 5 meters to the left you have to ask "According to which observer?".And since there is a conversion rate between proper distance and proper time ,time is also a dimension.And for the last 2 sentences you wrote down :this is according to the classical level.We dont know the nature of time in the quantum level which is what is fundamental $\endgroup$
    – Root Groves
    Commented Aug 14 at 8:08
  • $\begingroup$ 5 meters to the right means according to the observer bound to the matter "right here and right now". It's a statement that is only relevant in this one rest (mass) system. Take this local matter away and you are talking about nothing but a human abstraction that nature doesn't care about whatsoever. You are merely converting physicist's numbers in your head. There are a potential infinity of those everywhere and at all times and nature cares about none of them. $\endgroup$
    – FlatterMann
    Commented Aug 14 at 19:06

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