I'm having a bit of trouble with digesting an argument for which the conclusion is that it's not possible, in general, to Einstein synchronize clocks around closed curves. The argument goes as follows;(adsbygoogle = window.adsbygoogle || []).push({});

(Beginning)

Consider a four-velocity field ##e_{\hat{0}}## of the reference particles in a reference frame R.

In an arbitrary basis ##\{e_\mu\}## where we choose ##e_0## parallel to ##e_{\hat{0}}## the vectors ##e_i## need not be parallel to ##e_0##. We thus introduce

$$ e_{\perp i} = e_i - \frac{e_i \cdot e_0}{e_0 \cdot e_0} e_0 $$

and define

$$ \gamma_{ij} = e_{\perp i}\cdot e_{\perp j} = g_{ij} - \frac{g_{i0}g_{j0}}{g_00}$$

where ##g_{\mu \nu}## is the components of the metric tensor corresponding to the original general basis. Written in terms of the time orthogonal basis ##\{e_{\hat 0}, e_{\perp,i}\}## then we get

$$ds^2 = - d\hat t^2 + \gamma_{ij}dx^i dx^j.$$

We can thus write

$$ d\hat t^2 = \gamma_{ij}dx^i dx^j - ds^2 = \gamma_{ij}dx^i dx^j - g_{\mu \nu} dx^\mu dx^\nu = [(-g_{00})^{1/2}(dx^0 + \frac{g_{i0}}{g_{00}} dx^i)]^2.$$

Therefore ##d\hat t=0## corresponds to

$$ dx^0 = - \frac{g_{i0}}{g_{00}} dx^i$$

which is not a perfect differential. Thus the line integral of the coordinate time around a closed curve is in general not zero. The author then concludes that since ##d\hat t = 0## corresponds to simultaneity on Einstein synchronized clocks it is not in general (##g_{i0} \neq 0##) possible to Einstein synchronize clocks around closed curves.

(End)

Questions:

(1) Why does ##d\hat t = 0## correspond to Einstein synchronization of clocks?

(2) Is so bad that the line integral of the coordinate time is different from zero? Does this generally implies something other than a bad choice of time coordinate?

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Time synchronization in arbitrary reference frames.

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**