Is it Valid to Average Two Metrics in Spacetime?

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nomadreid
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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?
 
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  • #2
nomadreid said:
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
 
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  • #3
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.
 
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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.
 
  • #5
\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*}
$$
\overline{x}_{min}\;\leq\; \overline{x}_{harm} \;\leq\; \overline{x}_{geom} \;\leq\; \overline{x}_{arithm} \;\leq\; \overline{x}_{quadr}\;\leq\; \overline{x}_{cubic}\;\leq\; \overline{x}_{max}
$$
 
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  • #6
Super. Thanks, fresh_42.
 
  • #7
fresh_42 said:
\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)
 
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  • #8
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.
 
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  • #9
Good point, Ibix.
 
  • #10
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.
 
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  • #11
Super counter-example, martinbn! I shall keep it among my treasures. Thanks!
 

FAQ: Is it Valid to Average Two Metrics in Spacetime?

What does it mean to average two metrics in spacetime?

Averaging two metrics in spacetime refers to the process of combining two different spacetime geometries to create a single, averaged metric. This involves mathematical operations that take into account the curvature and other properties of each metric to produce a new metric that represents an average of the two original geometries.

Is it mathematically valid to average two metrics in spacetime?

Mathematically, averaging two metrics is a non-trivial task because metrics are tensor fields, and their averaging must preserve the geometric properties of spacetime. Simple arithmetic averaging is generally not valid as it can lead to a non-metric tensor. More sophisticated methods, such as Ricci flow or other geometric averaging techniques, are often required to ensure the result is a valid metric.

What are the challenges in averaging two spacetime metrics?

The main challenges include ensuring the averaged metric remains a valid solution to Einstein's field equations, preserving the differentiability and smoothness of the metric, and maintaining the physical and geometric properties of the original metrics. Additionally, there can be ambiguities in defining what constitutes an "average" in the context of curved spacetime.

Why might one want to average two metrics in spacetime?

Averaging two metrics can be useful in various contexts, such as simplifying complex spacetime geometries, studying perturbations around a known solution, or in cosmology where one might want to average over inhomogeneities in the universe to obtain a smoother, large-scale description of spacetime.

Are there any physical interpretations or applications of averaged spacetime metrics?

Yes, averaged spacetime metrics can have significant physical interpretations, especially in cosmology. For example, they can provide insights into the large-scale structure of the universe by averaging over small-scale fluctuations. They can also be used in numerical relativity to study the behavior of spacetime under various conditions, such as during the merger of black holes or in the presence of gravitational waves.

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