Question on linearity of Lorentz transformations

[facenian]

2. Clock rates are uniform, i.e. intervals measured by clocks agree with the linear structure of V.

what is the meaning of time intervals agreeing with the linear structure of V.

 

[Fredrik]

It does when b_i=0 for i=0,1,2,3

You also have to set the c_i=0.

 

[facenian]

You also have to set the c_i=0.

I don't see why. Setting b_i=0 for all i seems sufficient for T(0)=0

 

[Fredrik]

I don't see why. Setting b_i=0 for all i seems sufficient for T(0)=0

Your function isn't smooth, or even defined for all x, because of the x in the denominator.

 

[Fredrik]

I just started looking at the articles referenced on page 1. I haven't looked at George's reference yet (because it's not an online article), but all the others look interesting, especially the one atyy posted. It seems that the assumptions in my post #56 are much stronger than they need to be. I think I'm going to have to read that whole article soon.

 

[dx]

what is the meaning of time intervals agreeing with the linear structure of V?

If A - B = C - D, then a clock carried along AB will measure the same interval as it would if it were carried along CD. If A - B = λ(C - D), then a clock carried along AB will measure λ times the interval it would measure along CD.

 

[facenian]

If A - B = C - D, then a clock carried along AB will measure the same interval as it would if it were carried along CD. If A - B = λ(C - D), then a clock carried along AB will measure λ times the interval it would measure along CD.

Thank you dx. The fist condition seems to be consequence of homogeneity, and if A-B is understood only as distance could be consequence of isotropy.