Four-vector from 3 space components?

[jason12345]

If in some frame I define a geometrical object as having 3 space components, how do I then find the time component component it must have?

I've defined a geometrical object in some frame as having components Ax, Ay, Az defined by:

d/dx Ax = Fx, d/dx Ay = Fy, d/dy Ay = Fy

I require that for any frame:

1/gamma d/dx'A'x' = Fx, d/dy' A'y' = Fy, d/dy' A'y' = Fy

Is it possible to define the form the time component A0 takes?

Thanks.

 

[Fredrik]

Can't you just define your four-vector by specifying your three components in the frame in which the time component is zero, and then Lorentz transform to an arbitrary frame?

 

[pervect]

If you've mapped out a space grid with rulers, the usual approach is to put a bunch of clocks on that grid, synchronize them via the Einstein convention (assuming that this is possible, which requires the frame not be rotating), and then use the proper time read by each clock to determine the time coordinates of events.

I hope this answers your question?

 

[jason12345]

Can't you just define your four-vector by specifying your three components in the frame in which the time component is zero, and then Lorentz transform to an arbitrary frame?

I don't know what the time component is at all, so I can't say if it's zero or not, but you have given me an idea. I have (A0, A1, A2, A3), with A1' transforming as:

A1' = gamma (A1 - V A0)

Setting A1 = 0 for a suitable x,t gives:

A0 = - A1'/(gamma V) -- (1)

Cheers!