Lorentz Transformations: 2 Formulas for Time (t)

[len]

Depending on where I go to get a good understanding of the Lorentz transformations, I run into two formulas for time (t):

$$T=T_0 * \frac{1}{ \sqrt{ 1-\frac{v^2}{c^2} } }$$
and
$$t=\left( t' + \frac{vx'}{c^2} \right) * \frac{1}{ \sqrt{ 1-\frac{v^2}{c^2} } }$$

What is the explanation for having these two different formulas for time? If there was only one or the other, it would make sense to me but I can't understand how there can be two.

 

[ghwellsjr]

Depending on where I go to get a good understanding of the Lorentz transformations, I run into two formulas for time (t):

$$T=T_0 * \frac{1}{ \sqrt{ 1-\frac{v^2}{c^2} } }$$
and
$$t=\left( t' + \frac{vx'}{c^2} \right) * \frac{1}{ \sqrt{ 1-\frac{v^2}{c^2} } }$$

What is the explanation for having these two different formulas for time? If there was only one or the other, it would make sense to me but I can't understand how there can be two.

The first one is what you get when x'=0, in other words, for an object that is at rest at x'=0.

 

[Nugatory]

Depending on where I go to get a good understanding of the Lorentz transformations, I run into two formulas for time (t):

$$T=T_0 * \frac{1}{ \sqrt{ 1-\frac{v^2}{c^2} } }$$
and
$$t=\left( t' + \frac{vx'}{c^2} \right) * \frac{1}{ \sqrt{ 1-\frac{v^2}{c^2} } }$$

What is the explanation for having these two different formulas for time? If there was only one or the other, it would make sense to me but I can't understand how there can be two.

The second formula is the Lorentz transformation for the $t$ coordinate. It tells you how to calculate what time will appear on a clock in the unprimed frame when a clock in the primed frame reads a particular value.

The first formula is not a Lorentz transformation at all. It's the timedilation formula that tells you how how much time will have passed in the moving frame if a given amount of time has passed in the non-moving frame.