Length contraction, time dilation, and spacetime intervals contradiction

Question

I'm learning about spacetime and relativity in class. Default units are seconds and $c$. I'm using the length contraction calculator and time dilation calculator here: https://www.omnicalculator.com/physics/

But they don't seem to agree with spacetime interval invariants. I'm sure I'm making a noob mistake somewhere.

The spacetime interval for this frame is $0$.

$$x_0 = 10 \ \mathrm s \qquad t_0 = 10 \ \mathrm s \\ t_0 ^ 2 - x_0 ^ 2 = 0 $$

However, the spacetime interval for this frame is not $0$.

$$ v = 0.8 \qquad \gamma = 1.\bar{6} \\ x_1 = x_0 / \gamma = 6 \mathrm s \\ t_1 = t_0 \gamma = 16.\bar{6} \mathrm s \\ t_1 ^ 2 - x_1 ^ 2 = 241.\bar{7} $$

Answer 1

$x_1=x_0/γ$ and $t_1=t_0\cdot γ$ just aren't correct. You need to use the Lorentz transformation to find $x_1$ and $t_1$ in terms of $x_0$ and $t_0$.

The time dilation formula is a special case where $x_0=0$, which it isn't here, and the length contraction formula is a special case where $t_1=0$, which it isn't here.