Proper Time

1. Oct 1, 2008

Ben473

I know what the equation for proper time is in basic Euclaiden space. But when space-time is concerned, I get a bit confused.

The equation is: $$\Delta\tau=\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}$$

I realise that $$g_{\mu\nu}$$ is the Metric tensor. However i dont understand the dx's and their indices.

Would someone be able to explain these features to me?

Thanks,

Ben.

2. Oct 1, 2008

tiny-tim

Hi Ben!

It means dtau² is the linear combination of the dxidxjs, with coefficients gij.

So, for example, if gij is the usual (1,-1,-1,-1) diagonal tensor, then dtau² = dt² - dx² - dy² - dz².

3. Oct 1, 2008

Mentz114

The indexes on the dx's are tensor indexes, which run over the 4 dimensions so that for instance,
$$x^0 = t, x^1 = x, x^2 = y, x^3 = z$$

When indexes are repeated high and low, it means take the sum ( as Tiny-Tim has done ).

M

4. Oct 1, 2008

robphy

can be thought of as
$$\Delta\tau=\sqrt{ d\vec x \cdot d\vec x }$$
In the original form, the metric-tensor is explicit.

5. Oct 3, 2008

Ben473

Thanks Tiny Tim, Mentz 114 and Robphy.

Appreciate your help and I understand this equation alot better.

Thanks,

Ben.

P.S. Nice secret message Robphy!