In thegeneral theory of relativity, theMcVittie metric is the exact solution ofEinstein's field equations that describes ablack hole or massive object immersed in an expanding cosmologicalspacetime. The solution was first fully obtained byGeorge McVittie in the 1930s, while investigating the effect of the, then recently discovered,expansion of the Universe on a mass particle.

The simplest case of a spherically symmetric solution to thefield equations of General Relativity with acosmological constant term, theSchwarzschild-De Sitter spacetime, arises as a specific case of the McVittie metric, with positive 3-space scalar curvature${\displaystyle \kappa =+1}$ and constant Hubble parameter${\displaystyle H(t)=H_0}$.

Inisotropic coordinates, the McVittie metric is given by^[1]

${\displaystyle d{s}^2=-{\left(\frac{1-\frac{GM}{2c^{2}a(t)r}{K}^{1/2}(r)}{1+\frac{GM}{2c^{2}a(t)r}{K}^{1/2}(r)}\right)}^2{c}^{2}d{t}^2+\frac{{\left(1+\frac{GM}{2c^{2}a(t)r}{K}^{1/2}(r)\right)}^4}{K^2(r)}{a}^2(t)(d{r}^2+r^{2}d{\mathrm{\Omega }}^2),}$

where${\displaystyle d{\mathrm{\Omega }}^2}$ is the usualline element for the euclideansphere,${\displaystyle M}$ is identified as the mass of the massive object,${\displaystyle a(t)}$ is the usualscale factor found in theFLRW metric, which accounts for the expansion of the space-time; and${\displaystyle K(r)}$ is a curvature parameter related to thescalar curvature${\displaystyle \kappa }$ of the 3-space as

${\displaystyle K(r)=1+\kappa r^2=1+\frac{r^2}{4R^2},\kappa \in \{+1,0,-1\},}$

which is related to the curvature of the 3-space exactly as in theFLRW spacetime. It is generally assumed that${\displaystyle \dot{a}(t)>0}$, otherwise the Universe is undergoing acontraction.

One can define the time-dependent mass parameter${\displaystyle \mu (t)\equiv GM/2c^{2}a(t)r}$, which accounts for the mass density inside the expanding,comoving radius${\displaystyle a(t)r}$ at time${\displaystyle t}$, to write the metric in a more succinct way

${\displaystyle d{s}^2=-{\left(\frac{1-\mu (t)K^{1/2}(r)}{1+\mu (t)K^{1/2}(r)}\right)}^2{c}^{2}d{t}^2+\frac{{\left(1+\mu (t)K^{1/2}(r)\right)}^4}{K^2(r)}{a}^2(t)(d{r}^2+r^{2}d{\mathrm{\Omega }}^2),}$

From here on, it is useful to define${\displaystyle m=GM/c^2}$. For McVittie metrics with the general expanding FLRW solutions properties${\displaystyle H(t)=\dot{a}(t)/a(t)>0}$ and${\displaystyle \underset{t\to \mathrm{\infty }}{lim}H(t)=H_0=0}$, the spacetime has the property of containing at least twosingularities. One is a cosmological, null-likenaked singularity at the smallest positive root${\displaystyle r_{-}}$ of the equation${\displaystyle 1-2m/r-H_0^2{r}^2=0}$. This is interpreted as the black hole event-horizon in the case where${\displaystyle H_0>0}$.^[2] For the${\displaystyle H_0>0}$ case, there is an event horizon at${\displaystyle r=r_{-}}$, but no singularity, which is extinguished by the existence of an asymptotic Schwarzschild-De Sitter phase of the spacetime.^[2]

The second singularity lies at the causal past of all events in the space-time, and is a space-time singularity at${\displaystyle r=2m,\mu (t)=1}$, which, due to its causal past nature, is interpreted as the usualBig-Bang like singularity.

There are also at least two event horizons: one at the largest solution of${\displaystyle 1-2m/r-H_0^2{r}^2=0}$, and space-like, protecting the Big-Bang singularity at finite past time; and one at the${\displaystyle r=r_{-}}$ smallest root of the equation, also at finite time. The second event horizon becomes a black hole horizon for the${\displaystyle H_0>0}$ case.^[2]

One can obtain the Schwarzschild and Robertson-Walker metrics from the McVittie metric in the exact limits of${\displaystyle k=0,\dot{a}(t)=0}$ and${\displaystyle \mu (t)=0}$, respectively.In trying to describe the behavior of a mass particle in an expanding Universe, the original paper of McVittie a black hole spacetime with decreasing Schwarschild radius${\displaystyle r_s}$ for an expanding surrounding cosmological spacetime.^[1] However, one can also interpret, in the limit of a small mass parameter${\displaystyle \mu (t)}$, aperturbed FLRW spacetime, with${\displaystyle \mu }$ the Newtonian perturbation. Below we describe how to derive these analogies between the Schwarzschild and FLRW spacetimes from the McVittie metric.

In the case of a flat 3-space, with scalar curvature constant${\displaystyle k=0}$, the metric (1) becomes

${\displaystyle d{s}^2=-{\left(\frac{1-\frac{M}{2a(t)r}}{1+\frac{M}{2a(t)r}}\right)}^2{c}^{2}d{t}^2+{\left(1+\frac{M}{2a(t)r}\right)}^4{a}^2(t)(d{r}^2+r^{2}d{\mathrm{\Omega }}^2),}$

which, for each instant of cosmic time${\displaystyle t_0}$, is the metric of the region outside of aSchwarzschild black hole in isotropic coordinates, with Schwarzschild radius${\displaystyle r_S={\displaystyle \frac{2GM}{a(t_0)c^2}}}$.

To make this equivalence more explicit, one can make the change of radial variables

${\displaystyle r^{\prime }=r{\left(1+\frac{M}{2a(t)r}\right)}^2,}$

to obtain the metric inSchwarzschild coordinates:

${\displaystyle d{s}^2=-\left(1-\frac{2M}{a(t)r^{\prime }}\right)c^{2}d{t}^2+\left(1-\frac{2M}{a(t)r^{\prime }}\right)d{r}^{\prime 2}+r^{\prime 2}d{\mathrm{\Omega }}^2.}$

The interesting feature of this form of the metric is that one can clearly see that the Schwarzschild radius, which dictates at which distance from the center of the massive body theevent horizon is formed,shrinks as the Universe expands. For acomoving observer, which accompanies theHubble flow this effect is not perceptible, as its radial coordinate is given by${\displaystyle r_{(\text{comov})}^{\prime }=a(t)r^{\prime }}$, such that, for the comoving observer,${\displaystyle r_S=2M/r_{(\text{comov})}^{\prime }}$ is constant, and the Event Horizon will remain static.

In the case of a vanishing mass parameter${\displaystyle \mu (t)=0}$, the McVittie metric becomes exactly the FLRW metric in spherical coordinates

${\displaystyle d{s}^2=-c^{2}d{t}^2+\frac{a^2(t)}{{\left(1-\frac{r^2}{4R^2}\right)}^2}(d{r}^2+r^{2}d{\mathrm{\Omega }}^2),}$

which leads to the exactFriedmann equations for the evolution of the scale factor${\displaystyle a(t)}$.

If one takes the limit of the mass parameter${\displaystyle \mu (t)=M/2a(t)r\ll 1}$, the metric (1) becomes

${\displaystyle d{s}^2=-{\left(1-4\mu (t)K(r)\right)}^2{c}^{2}d{t}^2+\frac{\left(1+4\mu (t)K(r)\right)}{K^2(r)}{a}^2(t)(d{r}^2+r^{2}d{\mathrm{\Omega }}^2),}$

which can be mapped to a perturberd FLRW spacetime inNewtonian gauge, with perturbation potential${\displaystyle \mathrm{\Phi }=2\mu (t)}$; that is, one can understand the small mass of the central object as the perturbation in the FLRW metric.

• Cosmology
• Singularity theorems

• Exact solutions in general relativity
• Spacetime
• Mathematical methods in general relativity
• Lorentzian manifolds

• Articles with short description
• Short description is different from Wikidata