How can a tensor describe curvature?

grayson · December 20, 2023

Hello, I am trying to understand the Einstein Field Equations, but the only problem is that I cannot figure out how the einstein tensor describes the curvature of spacetime. There seems to be no good references or nifty little pictures showing the components like the one shown below for the stress-energy tensor.

image.png.ad946859ab7b9424964d04a4343432e9.png

I have seen one good thing that helps me understand some things, but it does not do much for explaining the components of the einstein tensor.

(Pdf shown below)

Pdf explaining the einstein field equations.

Merry Christmas.

KJW · December 20, 2023

The mathematical description of the Einstein tensor is actually quite complicated:

${\begin{aligned}G_{\alpha \beta }&=R_{\alpha \beta }-{\frac {1}{2}}g_{\alpha \beta }R\\&=R_{\alpha \beta }-{\frac {1}{2}}g_{\alpha \beta }g^{\gamma \zeta }R_{\gamma \zeta }\\&=\left(\delta _{\alpha }^{\gamma }\delta _{\beta }^{\zeta }-{\frac {1}{2}}g_{\alpha \beta }g^{\gamma \zeta }\right)R_{\gamma \zeta }\\&=\left(\delta _{\alpha }^{\gamma }\delta _{\beta }^{\zeta }-{\frac {1}{2}}g_{\alpha \beta }g^{\gamma \zeta }\right)\left(\Gamma ^{\epsilon }{}_{\gamma \zeta ,\epsilon }-\Gamma ^{\epsilon }{}_{\gamma \epsilon ,\zeta }+\Gamma ^{\epsilon }{}_{\epsilon \sigma }\Gamma ^{\sigma }{}_{\gamma \zeta }-\Gamma ^{\epsilon }{}_{\zeta \sigma }\Gamma ^{\sigma }{}_{\epsilon \gamma }\right),\\[2pt]G^{\alpha \beta }&=\left(g^{\alpha \gamma }g^{\beta \zeta }-{\frac {1}{2}}g^{\alpha \beta }g^{\gamma \zeta }\right)\left(\Gamma ^{\epsilon }{}_{\gamma \zeta ,\epsilon }-\Gamma ^{\epsilon }{}_{\gamma \epsilon ,\zeta }+\Gamma ^{\epsilon }{}_{\epsilon \sigma }\Gamma ^{\sigma }{}_{\gamma \zeta }-\Gamma ^{\epsilon }{}_{\zeta \sigma }\Gamma ^{\sigma }{}_{\epsilon \gamma }\right),\end{aligned}}$

where:

$\Gamma ^{\alpha }{}_{\beta \gamma }={\frac {1}{2}}g^{\alpha \epsilon }\left(g_{\beta \epsilon ,\gamma }+g_{\gamma \epsilon ,\beta }-g_{\beta \gamma ,\epsilon }\right).$

Actually, the above contains somewhat esoteric simplifying notation without which the expression would be even more complicated.

The above images were sourced from https://en.wikipedia.org/wiki/Einstein_tensor#Explicit_form

Edited December 20, 2023 by KJW

Genady · December 20, 2023

2 hours ago, grayson said:

how the einstein tensor describes the curvature of spacetime

Curvature is encoded in second derivatives of metric. So, any tensor that depends on second derivatives of metric, describes curvature in some way. Einstein tensor describes curvature in a way that can be related to the spacetime's physical contents.

studiot · December 20, 2023

39 minutes ago, Genady said:

Curvature is encoded in second derivatives of metric. So, any tensor that depends on second derivatives of metric, describes curvature in some way. Einstein tensor describes curvature in a way that can be related to the spacetime's physical contents.

+1

grayson · December 20, 2023

30 minutes ago, Genady said:

Curvature is encoded in second derivatives of metric. So, any tensor that depends on second derivatives of metric, describes curvature in some way. Einstein tensor describes curvature in a way that can be related to the spacetime's physical contents.

So, when you take the second derivative of the tensor, what do you get exactly? Is it the length that it bent or is it the length of the curvature itself?

Genady · December 20, 2023

23 minutes ago, grayson said:

So, when you take the second derivative of the tensor, what do you get exactly? Is it the length that it bent or is it the length of the curvature itself?

When you take second derivatives of metric, you get rates of its deviation from the flat spacetime.

Markus Hanke · December 21, 2023

12 hours ago, grayson said:

There seems to be no good references or nifty little pictures showing the components like the one shown below

I suspect what you’re looking for does not really exist, since, as others have pointed out, the Einstein tensor is actually quite a complicated function of the metric and its derivatives, so there’s no simple intuitive way to physically interpret any one of its components in isolation. But taken in its entirety, the tensor measures the extent by which geodesics will on average deviate around a small neighbourhood. It represents average Gaussian curvature in space once a time direction has been chosen.

The Einstein equations mean just this - average local curvature (of a very particular type) is precisely equivalent to local energy-momentum, up to a proportionality constant.

Note that this tensor only captures some aspects of total curvature, but it doesn’t provide a complete description (which is given by the Riemann tensor) - IOW, when you have G=0 then that does not necessarily mean you’re in a flat spacetime! In practice this tensor is seldom directly used in practical calculations - the Riemann tensor, Ricci tensor and Ricci scalar are more useful objects for practical purposes, and they have easier to understand physical meanings.

Sensei · December 21, 2023

In programming we call them data structures ("objects" in C++/Java). They can be pretty darn complicated; compared to them, a tensor looks like an offspring..

Data structures consist of simpler data structures that at their core consist of built-in types that are supported natively ("scalars").

A matrix is such a data type that is slightly after scalars and vectors and well below more complex data structures.

Edited December 21, 2023 by Sensei

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