The discussion centers on contracting the stress-energy tensor (SET) in the context of the Einstein field equations (EFE) and its implications for general relativity. Participants clarify that contracting the SET with the metric yields the Ricci scalar, specifically that the trace of the SET for a perfect fluid model is given by the expression \( T = -\epsilon + 3p \). The conversation also distinguishes between contractions using the metric and those using one-forms, emphasizing that while both yield invariant scalars, the latter can be coordinate-dependent. Key mathematical expressions and concepts are highlighted, including the importance of understanding tensor operations in the context of general relativity.
PREREQUISITESStudents and researchers in theoretical physics, particularly those focusing on general relativity, cosmology, and tensor analysis. This discussion is beneficial for anyone seeking to deepen their understanding of the mathematical framework underlying Einstein's theories.
Yes thanks for the clarification. But there are still units involved even though it's length over length I still consider those units because it relates to physical measurements.vanhees71 said:It's very convenient to normalize it to 1. The four-velocity of a massive particle is thus
$$u^{\mu}=\frac{1}{c} \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau},$$
which by definition of proper time is normalized to 1,
$$g_{\mu \nu} u^{\mu} u^{\nu}=1,$$
when using the west-coast convention for the pseudo-metric, i.e., the signature (1,-1,-1,-1).
Another even more convenient choice is to use natural units with ##\hbar=c=k_{\text{B}}=1## (and maybe even ##G=1##, which then makes everything measured in dimensionless quantities, i.e., Planck units).
dsaun777 said:Yes thanks for the clarification. But there are still units involved even though it's length over length I still consider those units because it relates to physical measurements.
If you are doing calculations on paper in an abstract manner to teach a class, I would say that you are not using any units. But if you are using applied physics to solve a real engineering problem it would be useful, I think, to include units even if they cancel and become dimensionless. If anything to show where unitless quantities get their respective derivation and meaning.robphy said:So, would you say the following?
In Euclidean geometry, the unit-vector along the x-direction \hat x is not dimensionless… but has units of m/m … or, if I were describing the direction of a force in classical physics, that unit vector along the force has units N/N or maybe N/\sqrt{N^2}.