The Ro a ing Disk Pa adox Is a T ans o ma ion E o
Max Ka son
No embe 20, 2025
Abs ac
S anda d ea men s o he Eh en es pa adox ely on he coo dina e ans o ma ion pos-
ula e o = lab. We demons a e ha his iden i y is physically in alid. Because spa ial
coo dina es a e de ined by ligh pa hs, he geome ic objec de ined as a ”disk” in he lab
ame maps o a non-ci cula geome y in he o a ing ame. The pa adox is no a physical
phenomenon bu a ans o ma ion e o a ising om an in alid assump ion ha he objec ’s
iden i y pe sis s ac oss ames.
The coo dina e e o . The ” o a ing disk pa adox” [1] is ypically de i ed by assuming ha
he adial coo dina e in he o a ing ame is iden ical o ha in he s a iona y lab ame ( = ′).
S anda d ea men s (e.g., G øn [2]) explici ly s ipula e his iden i y. This assump ion ea s he
disk as a New onian objec , a he han a geome ic measu emen de i ed om ligh pa hs. In
ela i i y, a spa ial dimension is de ined by simul aneous measu emen . Since simul anei y and
ligh p opaga ion pa hs di e be ween ine ial and o a ing ames, he se o e en s ha de ines a
s aigh adius in one ame does no de ine a s aigh adius in he o he . The iden i y = ′is
he e o e alse by de ini ion.
P oo ia mu ually exclusi e geome ies. Conside a ”disk” de ined in he lab ame: a se o
poin s simul aneously equidis an om a cen e , wi h s aigh lines h ough he cen e connec ing
an ipodal poin s. Fo an obse e on his o a ing objec ’s im, a pho on di ec ed adially inwa d
will miss he lab-de ined an ipode because i o a es away du ing ansi . Ope a ionally, he o a ing
obse e measu es a cho d, no a diame e . Thus, he objec de ined as a disk by he lab is de ined
as an asymme ic, non-ci cula shape by he o a ing obse e .
Conclusion. The pa adox a ises because he ans o ma ion = ′ o ces he o a ing obse e
o occupy he lab’s geome y while e aining hei own ime. This is impossible. The e is no single
”disk” ha exis s in bo h ames; he e a e wo dis inc geome ic de ini ions ha sha e no common
shape.
AI Disclosu e
The au ho used AI language models o assis wi h d a ing, de i a ions, and algeb aic checks.
Re e ences
[1] P. Eh en es , “Gleich ¨o mige Ro a ion s a e K¨o pe und Rela i i ¨a s heo ie,” Physikalische
Zei sch i 10, 918 (1909).
1
[2] Ø. G øn, “Rela i is ic desc ip ion o a o a ing disk,” Ame ican Jou nal o Physics 43, 869
(1975).
2
