# Why do we need the second postulate in special relativity? [duplicate]

• Postulate 1: All inertial frames are equivalent

• Postulate 2: The speed of light in vacuum has the same value $c$ in any inertial frame.

It can be shown that the Lorentz transformations directly follow from the fist postulate ( and by assuming the universe is isotropic, but this assumption is also needed with the standard derivation). This proof would lead to several cases to consider (namely, $c= \infty$, $0< c^2 < \infty$, $c^2=0$ and $c^2 <0$) however these can all be ruled out by experiment so that we finally reduce to the familiar Lorentz transformations. Why then do we still consider the invariance of the speed of light as a postulate, if it can be inferred from the first postulate?

## marked as duplicate by Alfred Centauri, Michael Seifert, stafusa, Chris♦, John Rennie special-relativity StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Mar 6 '18 at 11:30

• Using only the first postulate, we can derive relativity and that's cool. And then what shall we put for the value of $c$? Galilean relativity assumes $t=t'$ giving $c=\infty$whereas Einstein assumes $c=3\times 10^8 m/s$. – Yuzuriha Inori Mar 5 '18 at 18:10
• Possible duplicate of Einstein's first postulate implies the second? – John Rennie Mar 5 '18 at 18:12
• The Lorentz transformations involve an invariant speed $c$. That light propagates at the invariant speed $c$ is distinct from the existence of an invariant speed. – Alfred Centauri Mar 5 '18 at 18:13
• I don't think this is actually a duplicate. However, if I'm understanding the OP, then they haven't made it very explicit what they have in mind. I think they actually have in mind something along the lines of this paper: Palash B. Pal, "Nothing but Relativity," arxiv.org/abs/physics/0302045v1 . This approach goes back to Ignatowsky in 1911. The thing to understand is that there is more than one way of axiomatizing the same theory. There may be some overlap or similarity between Einstein's postulates and Pal's or Ignatowsky, but that doesn't mean one has to reduce to the other. – Ben Crowell Mar 6 '18 at 1:33

I think there's an analogy here with Newton's First Law of Motion. It's clearly a special case of the Second Law. But The First Law sets the scene, and that was an important thing to do in Newton's day.

In Einstein's time, c wasn't regarded as a space-time constant, a units conversion factor. Its only role was in electromagnetism, as the speed of e-m waves. The first postulate implied that c didn't depend on an observer's velocity; the second stopped up the loophole that it might depend on the source's velocity. I'm sure there's scope for re-formulating the postulates in different terms, taking the emphasis off e-m waves. Taylor and Wheeler went a long way towards doing this in their introductory book Spacetime Physics.

The first postulate was already known and accepted, before Einstein, yet they didn't have special relativity (instead the had classical relativity sometimes attributed to Galileo and sometimes to Newton).

Why not?

It helps to understand how scientists understood wave motion before relativity (and how will still understand the motion of all kinds of waves except light and gravitational waves). Wave speeds were understood as being relative a medium, and this has consequences. For instance, consider the difference between the treatment of the Doppler shift of sound (a wave in a medium) and that of light.

The second postulate says that the first postulate applies explicitly to the measured speed of light, which then re-casts all of classical physics.

• "The fist postulate was already known and accepted, before Einstein, yet they didn't have special relativity": Could it be argued that Einstein included electromagnetism, looking for the transformation that left Maxwell's equations invariant? – jim Mar 5 '18 at 18:47
• Well, yes. But the prevailing view was that EM waves would be like other waves: it would have a medium and $c = \sqrt{\epsilon_0 \mu_0}$ would be relative that medium, just like wavespeed on strings, and water surface and in air and so on. – dmckee Mar 5 '18 at 19:55