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Tahmeed
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Can a sphere or disk rotating with uniform speed follow born condition of rigidity?
Tahmeed said:Can a sphere or disk rotating with uniform speed follow born condition of rigidity?
pervect said:Yes.
stevendaryl said:Are you sure about that? I thought that Ehrenfest paradox showed that a rotating disk can't actually be perfectly rigid.
stevendaryl said:I thought that Ehrenfest paradox showed that a rotating disk can't actually be perfectly rigid.
Proposition 17.
Let u be a normalised timelike vector field
The motion described by its flow is rigid iff u is of vanishing shear and expansion
Vector fields generating rigid motions are now classified according to whether or not they have a vanishing vorticity ω : if
ω = 0 the flow is called irrotational, otherwise rotational. The following theorem is due to Herglotz and Noether:
Theorem 18 (Noether & Herglotz, part 1)
.
A rotational rigid motion in Minkowski space must be a Killing motion.
This motion corresponds to a rigid rotation with constant angular velocity κ
pervect said:A not particularly readable (IMO) reference on the topic is "The Rich Structure of Minkowskii Space" https://arxiv.org/abs/0802.4345. The notion of motion is described abstractly by a vector field ##u^a##, the integral curves of this vector field are the worldlines of particles on the body whose rigidity is to be tested. "Rigid motion" is a specific mathematical requirement that the vector field representing the motion must meet.
There's a couple of different ways of describing the mathematical requirements for the motion to be Born rigid. One of the more convenient ways is to say that the expansion and shear of the vector field u is zero. There's also an approach using Lie derivatives, which may be simpler in some respects. I believe from context that Born's original expression of the condition was in terms of the Lie derivative, for what it's worth.
The definition in terms of the expansion and shear is proposition 17 in the above reference:
Also of interest is the following:
Rotational Killing motioins do exist - they're just the motion of a rigid rotating disk. The paper even writes down the vector field of such a moton (using notation in which partial derivative operators are used to describe vectors) - it mentions that
Sorry if this is all too advanced, the complexities of a fuller explanation are why my first answer was just "Yes" and not more detailed.
Given that a sphere can rotate so as to satisfy Born's conditions any subset of the sphere will also satisfy Born's conditions. So the basic answer is yes.Tahmeed said:It is actually too advanced for me anyway, haha. However, does this only apply to spheres or anybody rotating uniformly?
pervect said:An assumption I make in giving this answer is that the space-time is the flat space-time of special relativity, not the curved space-time of General Relativity in the presence of significant mass. Rather than say this is necessary, I'd say that this is assumed.
The "born condition of rigidity" is a term used in science to describe the state of being born with a stiff or inflexible body. It can also refer to a genetic condition, such as osteogenesis imperfecta, that causes bones to be brittle and easily broken.
The exact cause of the "born condition of rigidity" can vary depending on the specific condition. In some cases, it may be due to genetic factors, while in others it may be caused by environmental influences during pregnancy or childbirth.
In some cases, the "born condition of rigidity" can be managed through various treatments such as physical therapy, medication, or surgery. However, there is currently no cure for genetic conditions that cause rigidity.
In some cases, the "born condition of rigidity" may be preventable by avoiding certain risk factors during pregnancy, such as smoking or alcohol consumption. However, genetic conditions that cause rigidity cannot be prevented.
The long-term effects of the "born condition of rigidity" can vary depending on the severity of the condition and the effectiveness of treatments. In some cases, individuals may experience difficulties with mobility and may require ongoing medical care. It is important for individuals with rigidity to work closely with their healthcare team to manage any potential long-term effects.