# Lorentz transformations formulas

1. Aug 5, 2011

### blueberrynerd

I'm slowly trying to understand sp relativity. I admit I got lost in the last thread I posted . But thanks to all who replied!

I have a question about the Lorentz transformations formulas. This is more of a mathematical question about how the formulas are derived.

If you have the two formulas,

x'= γ( x- vt) and x= γ(x' + vt')

which represent the x components for two reference frames S and S', and where γ is the Lorentz factor,

and you combine them to solve for t:

x'= γ[γ(x' + vt') - vt]

how do you arrive at the formula

t= γ(t' + vx'/c^2) ?

I know that you simply solve for t from the other formula, but I really cannot figure out how. Sorry, I realize this is more of a math-related problem, but I'm wondering if anybody can give me some tips?

2. Aug 5, 2011

### tiny-tim

hi blueberrynerd!

(try using the X2 icon just above the Reply box )
x'= γ[γ(x' + vt') - vt]

γvt = (γ2 - 1)x' + γvt'

t = (γ2 - 1)x'/γv + t'

(the https://www.physicsforums.com/library.php?do=view_item&itemid=19" is often easier to understand if you use the rapidity, α, defined by tanhα = v

then coshα = 1/√(1 - v2), sinhα = v/√(1 - v2), cosh2α - sinh2α = 1 )​

Last edited by a moderator: Apr 26, 2017
3. Aug 5, 2011

### Staff: Mentor

You also have to use the formula:
$$\gamma=\frac{1}{\sqrt{1-v^2/c^2}}$$

4. Aug 5, 2011

### grav-universe

Oops, just a small mistake there. It should be

x'= γ[γ(x' + vt') - vt]

γvt = (γ2 - 1)x' + γ^2 vt'

t = (γ2 - 1)x'/γv + yt'

Using y = 1 / sqrt(1 - (v/c)^2), it can then be reduced further to

t = (y - 1 / y) x' / v + y t'

t / y = (1 - 1 / y^2) x' / v + t'

t / y = (1 - (1 - (v/c)^2)) x' / v + t'

t / y = x' v / c^2 + t'

t = y (t' + x' v / c^2)

5. Aug 5, 2011

### GrayGhost

Blueberrynerd,

This may be the easiest way. You start with this ...

. x = γ(x' + vt')

and since x = ct, then ...

. x = γ(x' + vt')
. ct = γ(x' + vt')

and t' = x'/c, so ...

. ct = γ(x' + vt')
. ct = γ(x' + v(x'/c)

and x' = ct', so ...

. ct = γ(x' + vx'/c)
. ct = γ(ct' + vx'/c)

dividing thru by c ...

. t = γ(t' + vx'/c2))

GrayGhost

6. Aug 5, 2011

### DrGreg

Er, why should that be? The Lorentz transform applies to all values of x and t, not just the subset you consider.

7. Sep 21, 2011

### blueberrynerd

This reply is almost a month delayed, but THANK YOU! You helped a lot!

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