I have a question on the "natural metric" on the space of Bridgeland stability condition. 

A stability condition $\sigma=(Z,\mathcal{P})$ on a triangulated category $\mathcal{D}$ consists of a linear map $Z:K(\mathcal{D})\rightarrow \mathbb{C}$ called the central charge, and full additive subcacegories $\mathcal{P}(\phi) \subset \mathcal{D}$ for each $\phi \in \mathbb{R}$, satisfying the following four conditions:

 1. if non-zero $E \in \mathcal{P}(\phi)$, then we have $Z(E) = m_\sigma(E)exp(i\pi \phi)$ for some $m(E)\in \mathbb{R}_{+}$
 2. forall $\phi \in \mathbb{R}$,we have $\mathcal{P}(\phi+1)=\mathcal{P}(\phi)[1]$
 3. if $\phi_1 > \phi_2$ and $E_j \in \mathcal{P}(\phi_i)$ then $Hom_{\mathcal{D}}(E_1,E_2) = 0$
 4. for non-zero $E \in \mathcal{D}$ there exists a finite sequence of real numbers $\phi_1 >\phi_2 >\dots>\phi_n$ and $E$ obtained as an "iterated extension" of objects $A_i \in \mathcal{P}(\phi_i)$. 

We define $\phi^{+}_{\sigma(E)}=\phi_{1}$, $\phi^{-}_{\sigma(E)}=\phi_{n}$. The set of "loally finite" stablity conditions on $\mathcal{D}$ is denoted by $Stab(\mathcal{D})$. It has "a natural topology" induced by the following generalized metric
$$
d(\sigma_{1},\sigma_{2})=\sup_{0\ne E\in \mathcal{D}}\{|\phi^-_{\sigma_{1}}(E)-\phi^-_{\sigma_{2}}(E)|,|\phi^+_{\sigma_{1}}(E)-\phi^+_{\sigma_{2}}(E)|,|\log|\frac{m_{\sigma_{2}}(E)}{m_{\sigma_{1}(E)}}\}. 
$$
A celebrated result by Bridgeland says the forgetful map
$$
\mathcal{Z}:Stab(\mathcal{D})\longrightarrow Hom_{\mathbb{Z}}(K(\mathcal{D}),\mathbb{C})
$$
induces a local homeomorphism on each connected component of $Stab(\mathcal{D})$. This seems a really nice theorem.

This generalized metric is at this point beyond my intuition and I cannot really follow the proof of the theorem above, so let me now ask
> Why is the generalized metric above is "natural"? 

Of course some people may say it is the right one because the theorem holds. But I guess it is not the only reason. My problem is that I cannot really see why "distance" of two stability condition is measured by the only three quantities in $\sup_{0\ne E\in \mathcal{D}}$.