Definition of 4-vector quantities

[steveurkell]

Hi,
I just want to share my curiosity
in the definition of 4-vector quantities such as world line 4-vector x^alpha, 4-velocity vect, gauge potential etc. the ones with subscript for indices usually have the first component with negative sign and the ones with superscript for indices have all positive. For position coordinate x, as far as i know, the former is called covariant coordinate while the latter is called contravariant.
Do these covariant -contravariant terminology apply to other 4-vectors (velocity, energy momentum,etc)?
What are actually the differences between the two? I just doubt if the covariant corresponds to Minkowskian space while the contravariant to Euclidean space, is it correct?
Please correct me if there are wrong points in my statements. Thanks
Another my question, the relativistic momentum is p = gamma*m*v, m is rest mass
One is likely to say that the relativisticity of momentum is because the mass is relativistic, that is m' = gamma*m. Can we see this point from other perspective (though I am not quite sure if it is right), that the speed has been transformed to gamma*v while m is unchanged?
these must be simple questions for many of you
thanks for any help
regards

 

[pervect]

Covariant and contravariant apply to all 4 vectors. Here's the rationale behind the minus sign.

The length of a 4-vector is defined to be

$$\sum_{a=1}^{4} x^a x_a$$

Let's say our 4-vector is t,x,y,z. When you preform the sum with the sign conventions you've already described, you get

-t^2 + x^2 + y^2 + z^2

which is the Lorentz interval (with c=1 - the first comonet also gets multiplied by 'c' if you are not using geometric units with c=1)

Without the sign inversion, the length of the 4-vector wouldn't be its Lorentz interval.

In general one performs index lowering like this:
$$x_a = \sum_{a=1}^4 g_{ab} x^b$$

which is usually written in tensor notation without the sum being explicitly written out - tensor notation implicitly assumes that repeated indices where one index is raised and one is lowered are summed, thus we write only

$$x_a = g_{ab} x^b$$

Here g_00 = -1, g_11 = g_22 = g_33 = 1 represents the metric coefficients for a flat Minkowskian space-time.

g_ij is called the metric tensor, and as we've just seen it can be used to lower indices. It's matrix inverse, g^ij, is used to raise indices.

 

[dextercioby]

In classical electrodynamics & QFT,we adopt the other metric convention:

$$(\hat{g})_{\mu\nu}=\left( \begin{array}{cccc}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{array} \right)$$

for which the shorthand notation is

$$(\hat{g})_{\mu\nu}=\mbox{diag}(+,-,-,-)$$

In the flat limit of GR,indeed the convention & notation are different:

$$(\hat{\eta})_{\mu\nu}=\left( \begin{array}{cccc}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array} \right)$$

and a simple analogy with the I-st case would yield the shorthand notation.

Daniel.

 

[pmb_phy]

Hi,
I just want to share my curiosity in the definition of 4-vector quantities such as world line 4-vector x^alpha, 4-velocity vect, gauge potential etc.

A wordline is not a 4-vector. Examples that come to mind are the spacetime displacement 4-vector, The Lorentx 4-vector, the 4-vector of a four vector is the charge-current-4vector and the number 4-density f-vector. the ones with subscript for indices usually have the first component with negative sign and the ones with superscript for indices have all positive. For position coordinate x, as far as i know, the former is called covariant coordinate while the latter is called contravariant.

One is likely to say that the relativisticity of momentum is because the mass is relativistic, that is m' = gamma*m. Can we see this point from other perspective (though I am not quite sure if it is right), that the speed has been transformed to gamma*v while m is unchanged?
these must be simple questions for many of you
thanks for any help
regards

That a way of looking at things which cab provide weird interpretations. E.g. if I take the 4-velocity of a particle and dot it with Pete's 4-velocity then the redsult will be a tensor called a tensor of rank 0.

Now take the example of a type two tensor A(_,_) (A map of basis vectors , i.e. a basis set which maps tensors of rank 1 to scalars of rank two then the result is a tensor of rank 3. denote these basis vecotrs by e_1, e_2. Therefore A(e_1,e_2) is a scalar and is labeled A_{e_1,e_2)\

Pete