# Lorentz Transformations

I am trying to understand Lorentz Transformations in SR. I have some doubts.

1) In the derivation for Lorentz Transformations, why do we assume that Galilean Transformations hold true when an observer isn't in motion?

i.e. when you say $x'=\gamma (x-ut)$. You assume that an observer at rest, measures $x'=(x-ht)$.

2) Why do we think of multiplying only by a factor $\gamma$ and not adding or differentiating or some other linear operator to act on the measurements ?

3) Does the Second Postulate of SR, ultimately comes from the first one only i.e. all inertial observers are equal, hence there is no special inertial ether frame, thus speed of light must be same for all inertial observers. ( We already know the laws of electrodynamics are true)

## 1 Answer

You don't say how you're deriving the Lorentz transformations so it's a bit hard to comment.

The fundamental assumption in SR is that the proper time is an invarient i.e. if we have two events $(t, x, y, z)$ and $(t + dt, x + dx, y + dy, z + dz)$, so the interval between them is $(dt, dx, dy, dz)$, then the proper time defined by:

$$c^2d\tau^2 = c^2dt^2 - dx^2 - dy^2 - dz^2$$

will have the same value for all observers. All the weird effects like time dilation and length contraction as well as the factor of $\gamma$ originate from this assumption. Galilean invariance arises as a the zero velocity limit.

See for example my answer to How to calculate time dilation in approaching speed of light or search this site for proper time to find many other examples.

• youtube.com/watch?v=pHfFSQ6pLGU . Thank you for your answer. Here is the derivation. I do understand your answer partially, however, it is a bit advanced for me at the moment. Kindly see the derivation here.I'd be grateful if you can help in terms of that. – Isomorphic Dec 20 '13 at 8:48
• A 70 minute video!! I'm afraid you're going to have to condense the derivation here, showing which steps you have questions about. – John Rennie Dec 20 '13 at 9:20
• Please skip to the part in the description where it says lorentz transformation – Isomorphic Dec 20 '13 at 9:25
• And the time in the video for that section is ... – John Rennie Dec 20 '13 at 9:26
• Is there a need for assuming constancy of speed of light as a separate postulate along with the postulate that the space time metric is the same for all observers, or does the former follow from the latter? – Isomorphic Aug 16 '14 at 9:39