Understanding the Metric Tensor and its dx's in Space-Time

[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 realize that $$g_{\mu\nu}$$ is the Metric tensor. However i don't understand the dx's and their indices.

Would someone be able to explain these features to me?

Thanks,

Ben.

 

[tiny-tim]

$$\Delta\tau=\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}$$

I realize that $$g_{\mu\nu}$$ is the Metric tensor. However i don't understand the dx's and their indices.

Hi Ben!

It means dtau² is the linear combination of the dx_idx_js, with coefficients g_ij.

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

 

[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

 

[robphy]

$$\Delta\tau=\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}$$

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

 

[Ben473]

Thanks Tiny Tim, Mentz 114 and Robphy.

Appreciate your help and I understand this equation a lot better.

Thanks,

Ben.

P.S. Nice secret message Robphy!