Is it Valid to Average Two Metrics in Spacetime?

[nomadreid]

Due to the vagueness of this question, I am posting it in the Lounge, but if anyone suggests I clean it up and post it in a more specific forum, I will do so.
I came across a paper which, in itself, has no scientific value, but one passage in it piques my curiosity. The paper presents a couple of spacetime metrics, and then "averages" them. I have no idea whether this makes any sense. If it does, then just adding them and dividing by two end up with a proper metric? Is there a more valid way of averaging two metrics?

 

[Baluncore]

Is there a more valid way of averaging two metrics?

There are plenty to choose from.
Arithmetic mean. AM = (a+b) / 2
Harmonic mean. HM = 2/ ((1/a) + (1/b) )
Geometric mean. GM =√(a⋅b)
Arithmetic Geometric mean. AGM(a, b)
https://en.wikipedia.org/wiki/Arithmetic–geometric_mean

 

[Borek]

My understanding is that technically any combination of metrics that satisfies all four defining axioms is still a valid metrics. Averaging them by summing/dividing is definitely 'combining'.

Whether it adds anything to the general picture or to the case is another question.

 

[nomadreid]

Thanks, Baluncore and Borek. Of the four axioms for a metric, it appears that an average (whichever kind) would quickly satisfy the first three, but I suspect that the triangle inequality could conceivably pose difficulties in some cases. I will have to look at the article to see which metrics it combines, and how, and also play around with some metrics myself. Thanks for pointing me in the right direction.

 

[fresh_42]

\begin{align*}
\overline{x}_{min}&= \min\{\,x_1,\ldots,x_n\,\}&\text{ minimum }\\
\overline{x}_{harm}&= \dfrac{n}{\dfrac{1}{x_1}+\cdots+\dfrac{1}{x_n}}&\text{ harmonic mean }\\
\overline{x}_{geom}&= \sqrt[n]{x_1\cdot \cdots \cdot x_n}\; , \;x_k>0&\text{ geometric mean }\\
\overline{x}_{arithm}&= \dfrac{x_1+\cdots+x_n}{n}&\text{ arithmetic mean }\\
\overline{x}_{quadr}&= \sqrt{\dfrac{1}{n}\left(x_1^2+ \ldots + x_n^2\right)} &\text{ quadratic }\\
\overline{x}_{cubic}&= \sqrt[3]{\dfrac{1}{n}\left(x_1^3+ \ldots + x_n^3\right)} &\text{ cubic }\\
\overline{x}_{max}&= \max\{\,x_1,\ldots,x_n\,\} &\text{ maximum }\\
\end{align*}
$$
\overline{x}_{min}\;\leq\; \overline{x}_{harm} \;\leq\; \overline{x}_{geom} \;\leq\; \overline{x}_{arithm} \;\leq\; \overline{x}_{quadr}\;\leq\; \overline{x}_{cubic}\;\leq\; \overline{x}_{max}
$$

 

[nomadreid]

Super. Thanks, fresh_42.

 

[robphy]

\begin{align*}
\overline{x}_{min}&= \min\{\,x_1,\ldots,x_n\,\}&\text{ minimum }\\
\overline{x}_{harm}&= \dfrac{n}{\dfrac{1}{x_1}+\cdots+\dfrac{1}{x_n}}&\text{ harmonic mean }\\
\overline{x}_{geom}&= \sqrt[n]{x_1\cdot \cdots \cdot x_n}\; , \;x_k>0&\text{ geometric mean }\\
\overline{x}_{arithm}&= \dfrac{x_1+\cdots+x_n}{n}&\text{ arithmetic mean }\\
\overline{x}_{quadr}&= \sqrt{\dfrac{1}{n}\left(x_1^2+ \ldots + x_n^2\right)} &\text{ quadratic }\\
\overline{x}_{cubic}&= \sqrt[3]{\dfrac{1}{n}\left(x_1^3+ \ldots + x_n^3\right)} &\text{ cubic }\\
\overline{x}_{max}&= \max\{\,x_1,\ldots,x_n\,\} &\text{ maximum }\\
\end{align*}
$$
\overline{x}_{min}\;\leq\; \overline{x}_{harm} \;\leq\; \overline{x}_{geom} \;\leq\; \overline{x}_{arithm} \;\leq\; \overline{x}_{quadr}\;\leq\; \overline{x}_{cubic}\;\leq\; \overline{x}_{max}
$$

for "average velocity", we must include
\begin{align*}
\overline{v}_{avg}&=\dfrac{v_1 \Delta t_1+\cdots+v_n \Delta t_n }{\phantom{v_1}\Delta t_1 + \cdots +\phantom{v_n}\Delta t_n }&\text{ [time-weighted] average-velocity }\\
\end{align*}
(center of mass is another weighted-average)

 

[Ibix]

One point: essentially you can put any ten functions of four variables into the cells of a 4×4 symmetric matrix and call it a spacetime metric as long as it's invertible and has a Lorentzian signature. You can feed it through the field equations and get the stress-energy tensor you need to have that spacetime - which will usually have nothing physically plausible about it.

So it's more than likely that any vaguely reasonable combination of two metrics produces something you can call a metric. Whether it produces anything physically plausible or not is another matter.

It will also be a different spacetime from either of the contributing spacetime, so the thread title doesn't really make sense.

 

[nomadreid]

Good point, Ibix.

 

[martinbn]

For Riemannian metrics it is ok. For indefinite ones need not be the case. For example $g_1=-dt+dx+dy+dz$ and $g_2=dt-dx+dy+dz$. Their sum will be $dy+dz$, which is not Lorentzian.

 

[nomadreid]

Super counter-example, martinbn! I shall keep it among my treasures. Thanks!