Zero degree plus zero degree is how many kelvin? Addition in Affine spaces

Question

$0^{\circ} = 273.15$ K

If we consider the transformation which represents the degree to kelvin conversion, $T(x)= x +273.15$. Then it is bijective and can be considered as shifting up the line $y = x$ to the line $y = x + 273.15$, i.e. $K(x,y) = (x , y+275.15)$ for points $(x,y)$ in the line $x=y$. But this transformation is not linear, T$(0)\neq0, K(0,0)\neq(0,0)$, and in a way the structure is not preserved...

Question 1: Is that the reason why the following arguments won't work:

$273.15$ K $= 0^{\circ} = 0^{\circ} +0^{\circ}= 546.3$ K

And similarly, $(273.15 + 5)$ K $= 5^{\circ} = 2^{\circ} +3^{\circ}= (546.3 + 5)$ K

Question 2: Does that mean that translations of a set containing zero in the $X$-$Y$ plane can never be considered as homeomorphisms? Is there any structural property that is preserved in this translation?

How about in rigid body dynamics? Clearly, the euclidean distance is preserved. For eg: Distance between $(1,1)$ and $(2,2)$ is the same as the distance between $A(1,1)$ and $A(2,2)$ and so on...

Will addition be preserved under these homeomorphisms? Do we even need to define addition in the first place?

Answer 1

This is where all the problems start:

$0° \ C = 273.15 \ K \ $ ("is equal to")

Such expression are, in general, malformed as one cannot simply equate two different physical quantities (e.g. "$1 \ V = 1 \ J$"). See also the Wikipedia article on dimensional analysis, in particular this section, which also addresses differences between affine and vector quantities, including the example of temperature.

A correct statement would be

$0° \ C \ \widehat{=} \ 273.15 \ K \ $ ("corresponds to")

for absolute temperatures, or perhaps

$1° \ C = 1 \ K \ $ ("is equal to")

for relative temperatures, although I doubt that the latter is proper syntax as it is ambiguous. Usually, additional context or notation is required, e.g.

(the temperature is) at $1° \ C$

$T = 1° \ C$

vs.

(the temperature rises) by $1° \ C$

$\Delta T = 1° \ C$

which is why I would not recommend "mixing" (equating) different physical units, especially in isolated expressions. Also, this is entirely avoidable in most if not all cases.

PS.: I have seen usage of $\vartheta, \ \Delta \vartheta$ for temperatures on the Celsius scale and $T, \ \Delta T$ for temperatures on the Kelvin scale (the former mostly in chemistry and the latter mostly in physics literature), which would be another option for distinguishing between scales.

Answer 2

A temperature of $0^{\circ} C =$ a temperature of $273.15^{\circ} K$.

A temperature change of $0^{\circ} C = $ a temperature change of $0^{\circ} K$.

It's an affine space, as noted.

You can't add two temperatures and get another temperature - you can only add a temperature change to a temperature and get another temperature.

(Though you can add two temperature changes together and get another temperature change.)