Why isn’t it a minor nitpick? I mean, we use dimensioned constants in other areas; why, in principle, couldn’t the equation be E=mc (1 m/s)? If that was the only objection, and the theory made better predictions (which, obviously, it didn’t, but bear with me), then I don’t see any reason not to adopt it. Given that, I’m not sure why it should be a significant* objection.
Edited to add: Although I suppose that would privilege the meter and second (actually, the ratio between them) in a universal law, which would be very surprising. Just saying that there are trivial ways you can make the units check out, without tossing out the theory. Likewise, of course, the fact that the units do check out shouldn’t be taken too strongly in a theory’s favor. Not that anyone here hadn’t seen the XKCD, but I still need to link it, lest I lose my nerd license.
The whole point of dimensional analysis as a method of error checking is that fudging the units doesn’t work. If you have to use an arbitrary constant with no justification besides “making the units check out”, then that is a very bad sign.
If I say “you can measure speed by dividing force by area”, and you point out that that gives you a unit of pressure rather than speed, then I can’t just accuse you of nitpicking and say “well obviously you have to multiply by a constant of 1 m²s/kg”. You wouldn’t have to tell me why that operation isn’t allowed. I would have to explain why it’s justified.
Why isn’t it a minor nitpick? I mean, we use dimensioned constants in other areas; why, in principle, couldn’t the equation be E=mc (1 m/s)? If that was the only objection, and the theory made better predictions (which, obviously, it didn’t, but bear with me), then I don’t see any reason not to adopt it. Given that, I’m not sure why it should be a significant* objection.
Edited to add: Although I suppose that would privilege the meter and second (actually, the ratio between them) in a universal law, which would be very surprising. Just saying that there are trivial ways you can make the units check out, without tossing out the theory. Likewise, of course, the fact that the units do check out shouldn’t be taken too strongly in a theory’s favor. Not that anyone here hadn’t seen the XKCD, but I still need to link it, lest I lose my nerd license.
The whole point of dimensional analysis as a method of error checking is that fudging the units doesn’t work. If you have to use an arbitrary constant with no justification besides “making the units check out”, then that is a very bad sign.
If I say “you can measure speed by dividing force by area”, and you point out that that gives you a unit of pressure rather than speed, then I can’t just accuse you of nitpicking and say “well obviously you have to multiply by a constant of 1 m²s/kg”. You wouldn’t have to tell me why that operation isn’t allowed. I would have to explain why it’s justified.