Can the Lorentz Transformations be derived with only two conditions?

[rahuldandekar]

The book I use for relativity states that 4 conditions are required to get the four constants in the lorentz transformations. The 4 conditions the book uses are:
1) velocity of S' seen by S is v.
2) velocity of S as seen by S' is v.
3) Time dilation is same in either frame.
4) Speed of light is same in either frame.

However, my class uses Robert Resnick's "Introduction to Special Relativity" as the textbook. The derivation of the Lorentz transformations in that book is very different. Resnick uses the first condition as the velocity of S as seen by S'. However, the second condition used is that a sperical wave of light will remain spherical with speed c in both the frames. That equation is essentially x^2 = c^2*t^2 (in both frames), since y=y' and z=z'. This condition gives the three remaining constants.

But Resnick gets all 4 by using just the two conditions. Are the other two implicit in the second condition he uses? I am trying to wrap my head around this point, but I just cannot make sense of it.

 

[Meir Achuz]

I think you need (1), (2), and (4). (3) can then be derived.
Some derivations implicitly use (2) by saying that the two frames are equivalent.

 

[robphy]

Because of the numerous symmetries of Minkowski space,
there are *lots* of ways to get the Lorentz [boost] Transformations
...with varying starting points, levels of sophistication, and pedagogical strengths.

Here's a partial summary of approaches,
attached to an old post
https://www.physicsforums.com/attachment.php?attachmentid=4406&d=1122686537
( from "Spacetime and Electromagnetism" by J.R. Lucas, P.E. Hodgson )

For any "list of conditions" given in a derivation of the Lorentz boost transformations, there are likely other conditions that have been implictly assumed. Don't get too hung up on a particular derivation [unless you are willing to include the implicit assumptions in your study].