Center of Mass in Elliptical Space

[Ontophobe]

Apparently, a T-shaped organism in a positively curved space can wiggle around in such a way as to translate itself across space, thereby achieving locomotion simply by changing its shape in a specific series are ways. I've seen a computer generated animation of this and it's actually pretty cool. It's said that this is possible in elliptical space because the notion of "center of mass" is ill-defined in such a space.
What does this mean exactly? In what sense is the notion of "center of mass" not rigorously defined in curved space?

 

[andrewkirk]

A flat space is a vector space. That allows definition of the centre of mass of a system of particles in a coordinate-independent manner as

$$CoM=\frac{\sum_{i=1}^n m_i\mathbf{r}_i}{\sum_{i=1}^n m_i}$$

where ##\mathbf{r}_i## is the position vector of the ##i##th particle.

Unlike flat space, a curved space is not necessarily a vector space and, if it is not, the above definition is not available. One could make a definition based on coordinates instead, but that may make the CoM location vary by coordinate system.

 

[Ontophobe]

Thanks andrewkirk

Would you mind saying a little more about the difference between a vector space and a non-vector space, as such?

 

[Ontophobe]

Also, could we flesh out this concept of coordinate-dependency/independency a little more?

 

[andrewkirk]

Would you mind saying a little more about the difference between a vector space and a non-vector space, as such?

The key aspect of a vector space that is relevant here is that locations are things that can be added to one another, and that addition gives another point in the vector space. If we have the locations of two points we can add the vectors that represent those locations, divide by two, and get the midpoint between them. We can't do that if it's not a vector space. For instance, imagine if we are ants living on the surface of a balloon, and that surface is our world. If you use the above process to take the midpoint between two locations on the balloon, you get a place that's not on the balloon.

That's because, while the 3D space in which the balloon surface is embedded is a vector space, the balloon surface itself is not.

 

[andrewkirk]

Also, could we flesh out this concept of coordinate-dependency/independency a little more?

A coordinate-(in)dependent reference to a location is a reference that (doesn't) depend on some coordinate system to refer to the point. For instance if I tell you that the treasure is buried at a certain set of GPS coordinates, or at a certain latitude, longitude pair, I have given you a coordinate-dependent reference, and if your understanding of what those coordinates meant was different from mine (for instance if you thought that the zero meridian went through New York rather than London), you'd go to the wrong place. Alternatively, if I point at the ground and say 'dig there', or if I say 'It's buried underneath the pink palm tree' - and there's only one pink palm tree - then I have given you a coordinate-independent reference.