# Length contraction, time dilation, and spacetime intervals contradiction

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 $$0$$.

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

• Ty for editing my question. Jan 27, 2021 at 16:53

$$x_1=x_0/γ$$ 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.