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Ordinary Minkowski space is $\mathbb{R}^{3,1}:=(\mathbb{R}^4,\phi)$ where $\phi:\mathbb{R}^4\rightarrow\mathbb{R}$ is a quadratic form of signature $(3,1)$. Lying within this is a hyperboloid model for real hyperbolic 3-space $\mathbb{I}^3:=\{p=(w,x,y,z)\in\mathbb{R}^{3,1}\mid\phi(p)=-1\}/\{\pm 1\}$.

Consider replacing $\phi$ with a complex quadratic form $\psi:\mathbb{R}^{3,1}\rightarrow\mathbb{C}$, and consider a set of the form $S:=\{p=(w,x,y,z)\in\mathbb{R}^{3,1}\mid\psi(p)=-1, w>0\}$. What sort of shape does $S$ have? Are there choices for $a, b,c,d$ (besides $-1, 1, 1, 1$, respectively) that make $S/\{\pm1\}$ a hyperboloid model?

EDIT: I've posted the answer to the above question, so here is a follow-up question. Is there some insight to be gained from the cases discovered in the answer? Would something more enlightening happen if this were carried out in higher dimensions, i.e. taking a level set of a complex quadratic form on $\mathbb{R}^{n,1}$? That may be an imprecise question, but I wouldn't mind hearing opinions...

Ordinary Minkowski space is $\mathbb{R}^{3,1}:=(\mathbb{R}^4,\phi)$ where $\phi:\mathbb{R}^4\rightarrow\mathbb{R}$ is a quadratic form of signature $(3,1)$. Lying within this is a hyperboloid model for real hyperbolic 3-space $\mathbb{I}^3:=\{p=(w,x,y,z)\in\mathbb{R}^{3,1}\mid\phi(p)=-1\}/\{\pm 1\}$.

Consider replacing $\phi$ with a complex quadratic form $\psi:\mathbb{R}^{3,1}\rightarrow\mathbb{C}$, and consider a set of the form $S:=\{p=(w,x,y,z)\in\mathbb{R}^{3,1}\mid\psi(p)=-1, w>0\}$. What sort of shape does $S$ have? Are there choices for $a, b,c,d$ (besides $-1, 1, 1, 1$) that make $S/\{\pm1\}$ a hyperboloid model?

EDIT: I've posted the answer to the above question, so here is a follow-up question. Is there some insight to be gained from the cases discovered in the answer? Would something more enlightening happen if this were carried out in higher dimensions, i.e. taking a level set of a complex quadratic form on $\mathbb{R}^{n,1}$? That may be an imprecise question, but I wouldn't mind hearing opinions...

Ordinary Minkowski space is $\mathbb{R}^{3,1}:=(\mathbb{R}^4,\phi)$ where $\phi:\mathbb{R}^4\rightarrow\mathbb{R}$ is a quadratic form of signature $(3,1)$. Lying within this is a hyperboloid model for real hyperbolic 3-space $\mathbb{I}^3:=\{p=(w,x,y,z)\in\mathbb{R}^{3,1}\mid\phi(p)=-1\}/\{\pm 1\}$.

Consider replacing $\phi$ with a complex quadratic form $\psi:\mathbb{R}^{3,1}\rightarrow\mathbb{C}$, and consider a set of the form $S:=\{p=(w,x,y,z)\in\mathbb{R}^{3,1}\mid\psi(p)=-1, w>0\}$. What sort of shape does $S$ have? Are there choices for $a, b,c,d$ (besides something $-1, 1, 1, 1$, respectively) that make $S/\{\pm1\}$ a hyperboloid model?

EDIT: I've posted the answer to the above question, so here is a follow-up question. Is there some insight to be gained from the cases discovered in the answer? Would something more enlightening happen if this were carried out in higher dimensions, i.e. taking a level set of a complex quadratic form on $\mathbb{R}^{n,1}$? That may be an imprecise question, but I wouldn't mind hearing opinions...

Ordinary Minkowski space is $\mathbb{R}^{3,1}:=(\mathbb{R}^4,\phi)$ where $\phi:\mathbb{R}^4\rightarrow\mathbb{R}$ is a quadratic form of signature $(3,1)$. Lying within this is a hyperboloid model for real hyperbolic 3-space $\mathbb{I}^3:=\{p=(w,x,y,z)\in\mathbb{R}^{3,1}\mid\phi(p)=-1\}/\{\pm 1\}$.

Consider replacing $\phi$ with a complex quadratic form $\psi:\mathbb{R}^{3,1}\rightarrow\mathbb{C}$, and consider a set of the form $S:=\{p=(w,x,y,z)\in\mathbb{R}^{3,1}\mid\psi(p)=-1, w>0\}$. What sort of shape does $S$ have? Are there choices for $a, b,c,d$ (besides $-1, 1, 1, 1$, respectively) that make $S/\{\pm1\}$ a hyperboloid model?

EDIT: I've posted the answer to the above question, so here is a follow-up question. Is there some insight to be gained from the cases discovered in the answer? Would something more enlightening happen if this were carried out in higher dimensions, i.e. taking a level set of a complex quadratic form on $\mathbb{R}^{n,1}$? That may be an imprecise question, but I wouldn't mind hearing opinions...

I realize this question is weird, just looking for any intuition on this, or comparisons to existing concepts. (Or some reason why the idea is useless, would also be helpful!)

Ordinary Minkowski space is $\mathbb{R}^{3,1}:=(\mathbb{R}^4,\phi)$ where $\phi:\mathbb{R}^4\rightarrow\mathbb{R}$ is a quadratic form of signature $(3,1)$. Lying within this is a hyperboloid model for real hyperbolic 3-space $\mathbb{I}^3:=\{p=(w,x,y,z)\in\mathbb{R}^{3,1}\mid\phi(p)=-1\}/\{\pm 1\}$.

Consider replacing $\phi$ with a complex quadratic form $\psi:\mathbb{R}^{3,1}\rightarrow\mathbb{C}$, and consider a set of the form $S:=\{p=(w,x,y,z)\in\mathbb{R}^{3,1}\mid\psi(p)=-1, w>0\}$. What sort of shape does $S$ have? Are there choices for $a, b,c,d$ (besides something $-1, 1, 1, 1$, respectively) that make $S/\{\pm1\}$ a hyperboloid model?

Ordinary Minkowski space is $\mathbb{R}^{3,1}:=(\mathbb{R}^4,\phi)$ where $\phi:\mathbb{R}^4\rightarrow\mathbb{R}$ is a quadratic form of signature $(3,1)$. Lying within this is a hyperboloid model for real hyperbolic 3-space $\mathbb{I}^3:=\{p=(w,x,y,z)\in\mathbb{R}^{3,1}\mid\phi(p)=-1\}/\{\pm 1\}$.

Consider replacing $\phi$ with a complex quadratic form $\psi:\mathbb{R}^{3,1}\rightarrow\mathbb{C}$, and consider a set of the form $S:=\{p=(w,x,y,z)\in\mathbb{R}^{3,1}\mid\psi(p)=-1, w>0\}$. What sort of shape does $S$ have? Are there choices for $a, b,c,d$ (besides something $-1, 1, 1, 1$, respectively) that make $S/\{\pm1\}$ a hyperboloid model?

EDIT: I've posted the answer to the above question, so here is a follow-up question. Is there some insight to be gained from the cases discovered in the answer? Would something more enlightening happen if this were carried out in higher dimensions, i.e. taking a level set of a complex quadratic form on $\mathbb{R}^{n,1}$? That may be an imprecise question, but I wouldn't mind hearing opinions...