Introduction

To better understand and visualize the underlying geometry in a curved spacetime one could find and employ its two-dimensional embedding diagrams. The well-known Flamm’s paraboloid1 gives a snapshot of the equatorial Schwarzschild spacetime embedded in a three-dimensional Euclidean space. This helps to visualize both the spatial geometry represented by this solution, and its time-projected geodesics (particle orbits). There are also the dynamic or spacetime embedding diagrams introduced in2,3, which embed the (t, r)-surface (radial plane) of a curved spacetime in a three-dimensional Minkowski spacetime.

Here we will discuss both types of the embeddings, the spatial and the dynamic ones corresponding to the equatorial surfaces, and radial planes in stationary spacetimes respectively. Obviously the embedding diagrams of stationary spacetimes are expected to be different from those given for static spacetimes. Two vacuum stationary spacetimes which are somewhat direct generalizations of the Schwarzschild solution are the Kerr4, and NUT5 spacetimes. Both are axially symmetric spacetimes with two parameters, one of them the mass parameter. The Kerr solution interpreted as the spacetime of a rotating mass includes the angular momentum per unit mass as the second parameter, while the NUT solution which is interpreted as the gravitational analog of a magnetic monopole, or the so called gravitomagnetic monopole, includes the NUT parameter also called the magnetic mass, as the second parameter.

The well-known Kerr solution in Boyer-Lindquist coordinates6 is given by the metric,

$$\begin{aligned} d{s^2} = \left( 1 - \frac{{2mr}}{{{\rho ^2}}}\right) d{t^2} + \frac{{4mar{{\sin }^2}\theta }}{{{\rho ^2}}}dtd\phi - \frac{{{\rho ^2}}}{\Delta }d{r^2} - {\rho ^2}d{\theta ^2} - \left( {r^2} + {a^2} + \frac{{2m{a^2}r{{\sin }^2}\theta }}{{{\rho ^2}}}\right) {\sin ^2}\theta d{\phi ^2} \end{aligned}$$
(1)

with

$$\begin{aligned} {\rho ^2} = {r^2} + {a^2}{{\cos }^2}\theta \;\;, \;\;\; \Delta = {r^2} - 2mr + {a^2}. \end{aligned}$$
(2)

in which m and a are the mass and angular momentum per unit mass respectively. For \(a=0\) it reduces to Schwarzschild solution, and for \(m=0\), and \(r\ge 0\) to flat spacetime in oblate spheroidal coordinates (the zero mass limit of the extended (\(-\infty< r < \infty\)) Kerr spacetime is discussed in7). For \(m^2 > a^2\) it has two horizons (outer and inner) with radii

$$\begin{aligned} r_{\pm } = m \pm \sqrt{m^2-a^2} \end{aligned}$$
(3)

and a ring singularity of radius a lying in the equatorial plane behind its horizons8.

NUT spacetime on the other hand has a number of exotic, and at the same time interesting properties. Its metric in Schwarzschild-type coordinate is given by

$$\begin{aligned} ds^2 = f(r) (dt - 2l \cos \theta d\phi )^2 - \frac{dr^2}{f(r)} - (r^2 + l^2) d\Omega ^2 \end{aligned}$$
(4)

with

$$\begin{aligned} f(r) = \frac{r^2 - l^2 - 2mr}{r^2 + l^2}, \end{aligned}$$
(5)

in which m and l are the mass, and NUT parameters respectively. For \(l=0\) we recover the Schwarzschild metric, whereas for \(m=0\), unlike the Kerr case, we do not find flat spacetime in an exotic coordinate, but the so called pure NUT spacetime which is a one-parameter stationary spacetime. Taking the radial coordinate \(0< r < \infty\), there is a coordinate singularity at \(r_H = m + (m^2 + l^2)^{1/2}\) (where \(f(r)=0\)) representing the NUT horizon, which unlike the horizons of Schwarzschild and Kerr black holes, is not hiding an intrinsic singularity. It is noted that as in the case of Schwarzschild spacetime inside the horizon, i.e for \(0< r < r_H\) where \(f(r) < 0\), r and t change their role. This region is called the Taub region, and its metric which is time-dependent is the so called Taub metric (solution) discovered before the NUT solution9. This is why the whole spacetime, including both the time-dependent and stationary sectors, is called the Taub-NUT spacetime.

Along the z-axis (\(\theta =0, \pi\) ) the NUT spacetime has a string singularity10,11,12 which in comparison with a Dirac monopole in electromagnetism, and its string singularity, allows an interpretation of NUT spacetime as the spacetime of a mass endowed with a gravitomagnetic monopole charge (the NUT parameter l)13. This justifies its physical spherical symmetry despite its mathematically axisymmetric appearance14,15. Through a coordinate transformation, Bonnor reduced the singularity to half-axis, giving the NUT spacetime another interpretation in terms of a central mass plus a semi-infinite massless source of angular momentum16. Another interesting feature of NUT spacetime, also originated from its Dirac-type string singularity, is the fact that it is locally but not globally asymptotically flat. To put it another way, its Riemann tensor vanishes as \(r\rightarrow \infty\), but its metric reduces to that of flat spacetime except in the direction of the singularity.

Thermodynamics of Taub-NUT spacetime in four and higher dimensions17, as well as their extended versions including other pramameters such as rotation, charge and cosmological constant have been the focus of many studies18,19,20,21,22,23. Taub-NUT spacetime also played an important role in string theory. Indeed it was first embedded as a nontrivial stringy solution in the so called heterotic string theory24.

Observationally, the NUT spacetime, as the spacetime around an astrophysical object, could be identified from its distinct effects on light rays as a gravitational lens25,26. For more detail on Taub-NUT spacetime, and its characteristics refer to the appendix (Supplemental Material to this article).

In what follows we will begin with the spatial embedding diagram of the equatorial NUT spacetime in three dimensional Euclidean space, by restricting the r-coordinate to the stationary region outside the horizon \(r_H\). This can not be done analytically, so by reducing the embedding equation to the cases of Schwarzschild and pure NUT spaces, we show that one can find, in both cases, the embedding diagrams both analytically and numerically which agree perfectly. This provides an evidence for the reliability of our numerical calculations for the embedding diagrams of Kerr and NUT spacetimes. Next we calculate the Gaussian and mean curvatures of the embedding surfaces, and to better compare all these embedding diagrams, and their curvatures as a function of the embedding radial coordinate, we plot them in a single diagram by fixing their corresponding horizons at a given common value.

Finally we will discuss the dynamic embedding diagram of NUT spacetime which includes the region inside the horizon, and compare its diagram with the corresponding diagram given for the Schwarzschild spacetime.

Embedding diagram of equatorial NUT spacetime

A time section (\(t=constant\)) of the NUT spacetime in the equatorial plane (\(\theta = \frac{\pi }{2}\)) is given by the following 2-dimensional metric.

$$\begin{aligned} ds^2 = \frac{dr^2}{f(r)} + (r^2 + l^2) d\phi ^2 \end{aligned}$$
(6)

Obviously, taking the string singularity along the z-axis, this is a two-dimensional asymptotically flat space. As in the case of Schwarzschild spacetime, to better visualize the geometry of the above 2-surface we need to embed it into a three dimensional Euclidean space. To do so first we take \((r^2 + l^2) = R^2\), so that the above metric reduces to

$$\begin{aligned} ds^2 = \frac{R^4}{(R^2 - l^2)(R^2 - 2l^2 - 2m(R^2 - l^2)^{1/2})} dR^2 + R^2 d\phi ^2 \end{aligned}$$
(7)

Now using the standard procedure we can embed this surface in a three-dimensional Euclidean space with the following metric

$$\begin{aligned} ds^2 =dZ^2 + dR^2 + R^2 d\phi ^2 \end{aligned}$$
(8)

written in the following form

$$\begin{aligned} ds^2 =(1 + \frac{dZ^2}{dR^2})dR^2 + R^2 d\phi ^2. \end{aligned}$$
(9)

So that comparing the above equation with (7), we end up with the following integral for \(Z=Z(R)\)

$$\begin{aligned} Z(R) = \int \left( \frac{3l^2 R^2 + 2m(R^2-l^2) ^{3/2} -2l^4}{(R^2-l^2)[R^2-2l^2 - 2m(R^2 - l^2)^{1/2}]}\right) ^{1/2} dR. \end{aligned}$$
(10)

in which the result of integration is valid for \(R > R_H =({r_H}^2 + l^2)^{1/2}\). Now there are different cases that one can consider:

(I):

The case of Schwarzschild spacetime (\(l=0\)) : In this well known case, where \(R=r\), the above integral leads to the following curve

$$\begin{aligned} Z^2 =8m(r-2m) \end{aligned}$$
(11)

whose surface of revolution about the Z-axis, valid for \(r > r_H = 2m\), is the well-known Flamm’s paraboloid. In Fig. 1 the above analytical curve and the numerical integration of Eq. (10) for \(l=0\) are shown on the same plot which agree perfectly. This could be taken as a test of validity and precision of the employed numerical integration.

(II):

The case of pure NUT spacetime (\(m=0\)): Here we have the following integral valid for \(R > R_H = \sqrt{2}l\),

$$\begin{aligned} Z(R) = \int \left( \frac{3l^2 R^2 - 2l^4}{(R^2-l^2)(R^2-2l^2)}\right) ^{1/2} dR. \end{aligned}$$
(12)
Figure 1
figure 1

Flamm’s parabola in Schwarzscild spacetime with \(m=1\).

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Figure 2
figure 2

Flamm’s paraboloid in Schwarzschild spacetime.

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Figure 3
figure 3

Analog of Flamm’s parabola in pure NUT spacetime with \(l=1\).

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Figure 4
figure 4

Analog of Flamm’s paraboloid in pure NUT spacetime with \(l=1\).

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which interestingly enough could be calculated in terms of elliptic integrals through the change of variable \(u = -2 l^4-3 l^2 R^2\), leading to the following curve

$$\begin{aligned} Z(R) =\pm \frac{1}{2} l \left( F\left( \csc ^{-1}\left( \sqrt{\frac{2 l^2}{R^2-2 l^2}+2}\right) |\frac{1}{2}\right) +3 \Pi \left( 2;\csc ^{-1}\left( \sqrt{\frac{2 l^2}{R^2-2 l^2}+2}\right) |\frac{1}{2}\right) \right) \end{aligned}$$
(13)

where

$$\begin{aligned} \Pi (n;\phi |m)=\int _0^{\phi }d \theta \frac{1}{\sqrt{1-m \sin ^2(\theta )} \left( 1-n \sin ^2(\theta )\right) } \end{aligned}$$
(14)

is the incomplete elliptic integral of the third kind, and

$$\begin{aligned} F(\phi |m)=\int _0^{\phi } \frac{1}{\sqrt{1-m \sin ^2(\theta )}} \, d\theta \end{aligned}$$
(15)

is the incomplete elliptic integral of the first kind. Having the values of these integrals, the embedding curve of pure NUT is depicted, along with the direct numerical integration, in Fig. 3. Again the two results coincide perfectly. The surface revolution of the curves in Figs. 1 and 3, corresponding to Flamm’s paraboloid and its analog in pure NUT spacetime, are shown in Figs. 2 and 4 respectively. From Figs. 1 and 3 it is obvious that the embedding geometry in the case of pure NUT spacetime, as compared to Flamm’s parabola, starts flattening at shorter embedding radii.

Figure 5
figure 5

Embedding diagram of NUT spacetime with two different sets of values for m and l, but the same embedding horizon radius \(R_H = 1\).

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(III):

The general case of NUT spacetime with non-zero mass and NUT parameter: In this case the integral (Eq. 10) can not be performed analytically or in terms of known functions such as elliptic integrals, and so we employ numerical integration to obtain the embedding diagram. For example the resulted embedding curves for two different sets of values for m and l with the same embedding horizon radius \(R_H = 1\), are shown in Fig. 5. Comparing with Figs. 1 and 3, this plot shows how the larger values of m or l dominate the asymptotic behaviour of the embedding curve in approaching to those corresponding to Schwarzschild and pure NUT spacetimes respectively. Below we will see that this same pattern is repeated in the behaviour of the Gaussian and mean curvatures of the NUT embedding diagrams.

Before doing so it should be pointed out that in a recent study27 metamaterial analog of closed photon orbits, the so called photon rings, in the equatorial NUT spacetime are simulated. These exact ray-tracing simulations are based on the metamaterial’s index of refraction which is adapted from the corresponding spacetime index of refraction28. Having the above embedding diagrams of NUT spacetime helps us to visualize these photon rings as well as other null and timelike orbits in NUT spacetime. Indeed they are the time-projected photon orbits on the above introduced embedding diagram. As in the case of Schwarzschild spacetime29, it is expected that the spatial curvature of NUT spacetime, manifested in its embedding diagram, to be partially responsible for light bending around a NUT hole25.

Gaussian and mean curvatures of NUT embedding diagrams

Another way that one could compare the spatial embedding diagrams of NUT, Schwarzschild and pure NUT spacetimes, is through their Gaussian and mean curvatures. Obviously since these two dimensional surfaces only depend on the embedding radial coordinate R, so will be their Gaussian and mean curvatures. The general formulae for the Gaussian and mean curvatures of these surfaces are given by30

$$\begin{aligned} K = \frac{Z' Z''}{R \left( 1+Z'^2\right) ^2} \end{aligned}$$
(16)
$$\begin{aligned} H = \frac{R Z''+Z'^3+Z'}{2 R \left( 1+Z'^2\right) ^{3/2}} \end{aligned}$$
(17)

where \(`` \prime ``\) denotes differentiation with respect to the embedding radial coordinate R. For the NUT family one finds

$$\begin{aligned} K_{NUT} = \frac{4 l^4+l^2 \left( 4 m \sqrt{R^2-l^2}-3 R^2\right) -m R^2 \sqrt{R^2-l^2}}{R^6} \end{aligned}$$
(18)
$$\begin{aligned} H_{NUT} = \frac{2 l^4+\sqrt{R^2-l^2} \left( 2 l^2 m+m R^2\right) }{2 R^3 \sqrt{l^4+2 m \left( R^2-l^2\right) ^{3/2}+3 l^2 \left( R^2-l^2\right) }} \end{aligned}$$
(19)

which for \(l=0\), and \(m=0\) reduce to the following values

$$\begin{aligned} K_{Sch} = -\frac{m}{R^3}\;\;\;\;\;\;\;; \;\;\;\;\;\;\;\; H_{Sch} = \frac{1}{2R}\sqrt{\frac{m}{2R}} \end{aligned}$$
(20)

and

$$\begin{aligned} K_{PNUT} = \frac{4 l^4-3 l^2 R^2}{R^6}\;\;\;\;\;\;\;; \;\;\;\;\;\;\;\; H_{PNUT} =\frac{l^3}{R^3 \sqrt{3 R^2-2 l^2}} \end{aligned}$$
(21)

respectively. Obviously the NUT spacetime embedding curvatures (18) and (19) reduce to their limiting values (20) and (21), when \(\frac{l}{R} \ll 1\) and \(\frac{m}{R} \ll 1\), respectively. Indeed these are the same limiting values shown in the two diagrams in Fig. 5.

Embedding diagrams of Reissner-Nördstrom and Kerr spacetimes

Although Reissner–Nördstrom (R-N) spacetime is a static spacetime, but having two parameters makes it a good example to compare its embedding diagram with that of NUT spacetime we examined in the last section, and the Kerr spacetime which we will discuss later on in this section.

Embedding diagram of equatorial R–N spacetime

Embedding diagrams for R–N spacetime, both spatial and dynamic, have already discussed in the literature (see31,32). The R–N spacetime as a static spherically symmetric solution of the Einstein–Maxwell equations, is given by the following line element

$$\begin{aligned} ds^{2}= & {} f(r)dt^2 - \frac{dr^2}{f(r)} - r^2d\Omega ^2 \end{aligned}$$
(22)
$$\begin{aligned} f(r)= & {} 1 - \frac{2m}{r} + \frac{q^2}{r^2} \end{aligned}$$
(23)

in which the two parameters are mass m and charge q. It reduces to the Schwarzschild spacetime for \(q=0\), and has two horizons at \(r_{\pm } = m \pm \sqrt{m^2-q^2}\). At a given instant of time and in the equatorial plane we have

$$\begin{aligned} ds^{2} = \frac{dr^2}{f(r)}+r^2d\phi ^2. \end{aligned}$$
(24)

Noting that here, as in the case of Schwarzschild spacetime, \(R=r\), and employing the standard embedding procedure, the equatorial R-N embedded in Euclidean space is given by the curve \(Z(R)\equiv Z(r)\) through the following integral,

$$\begin{aligned} Z(r) = \int \sqrt{\frac{2mr - q^2}{r^2 - 2mr + q^2}}dr \end{aligned}$$
(25)

valid for \(r > r_+\), leading to

$$\begin{aligned} Z(r)=\frac{2 \left( E\left( \frac{4 m \sqrt{m^2-q^2}}{q^2+2 m \left( \sqrt{m^2-q^2}-m\right) }\right) -E\left( \csc ^{-1}\left( \frac{\sqrt{2} \root 4 \of {m^2-q^2}}{\sqrt{-m+r+\sqrt{m^2-q^2}}}\right) |\frac{4 m \sqrt{m^2-q^2}}{q^2+2 m \left( \sqrt{m^2-q^2}-m\right) }\right) \right) }{\sqrt{\frac{1}{2 m \left( \sqrt{m^2-q^2}-m\right) +q^2}}} \end{aligned}$$
(26)

where the elliptic integral E is defined as follows

$$\begin{aligned} E(\phi |m)=\int ^\phi _0 (1-m\sin ^2\theta )^{\frac{1}{2}} d\theta \;\;\;; \;\;\; E(m)\equiv E(\pi /2|m). \end{aligned}$$
(27)

The embedding diagram of R-N spacetime, based on (26) is shown in Fig. 6.

Figure 6
figure 6

Analog of Flamm’s parabola for R-N spacetime with \(m=2\) and \(q=1\).

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Using the formulas (16) and (17), the Gaussian and mean curvatures of embedding diagrams of R-N are given by

$$\begin{aligned} K_{R-N} = \frac{q^2-m R}{R^4}\;\;\;\;; \;\;\;\; H_{R-N} = \frac{m}{2 R \sqrt{2 m R-q^2}} \end{aligned}$$
(28)

Embedding diagram of equatorial Kerr metric

Embedding diagrams for the equatorial kerr metric were first discussed in33, in which the authors gave a schematic illustration of these diagrams for near-extreme Kerr black holes. Also embedding diagrams in Kerr-Newmann optical reference geometry are discussed in34.

Kerr metric in the equatorial plane, and at an instant of time reduces to

$$\begin{aligned} ds^{2} = \frac{r^2}{r^2-2mr+a^2}dr^2+\left( r^2+a^2+\frac{2ma^2}{r}\right) d\phi ^2. \end{aligned}$$
(29)

Now taking \(R^2 = r^2+a^2+\frac{2ma^2}{r}\), from (3), the outer horizon will be at \(R_H = 2m\). In other words \(R_H\) in the Kerr case is the same as in the Schwarzschild metric, so unlike the previous cases there is no trace of the second parameter, (the angular momentum a) in the horizon radius given in terms of the embedding radial coordinate.

To find r(R) we need to solve the cubic equation \(r^3+r(a^2-R^2) + 2ma^2 =0\) with discriminant \(D= (\frac{a^2-R^2}{3})^{3} + m^2a^4\). Taking into account that \(R \ge 2m\), and \(a\le m\), the largest value of D, which is zero, is given by \(R = 2m\), and \(a=m\). In other words we always have \(D\le 0\) with the equal sign holding for an extreme Kerr black hole. In the general case \(D<0\), one can show that the only solution which reduces to \(r=R\) for \(a=0\) is given by

$$\begin{aligned} r= \frac{2}{\sqrt{3}} \sqrt{R^2-a^2} \cos \left( \frac{1}{3} \cos ^{-1}\left( -\frac{3 \sqrt{3} a^2 m}{\left( R^2-a^2\right) ^{3/2}}\right) \right) . \end{aligned}$$
(30)

Now applying the standard embedding procedure we end up with the following equation for Z(R) to be solved for the embedding diagram

$$\begin{aligned} \frac{dZ}{dR}= \sqrt{\frac{m \left( -a^6 m+a^4 r \left( 2 m^2-m r+2 r^2\right) +4 a^2 r^4 (r-m)+2 r^7\right) }{\left( r^3-a^2 m\right) ^2 \left( a^2+r (r-2 m)\right) }} \end{aligned}$$
(31)

The results of numerical integration of Eq. (31), valid for \(R > R_H = 2m\), gives the embedding diagrams in Fig. 7. These are for two different sets of values for m and a, with the same \(R_H = 1\). It is seen that when the value of a is very close to m, as in the case of extreme Kerr black hole, the embedding curve tends to become flat with a less steeper slope.

Figure 7
figure 7

Embedding curve for Kerr spacetime with two different sets of values for m and a, but the same embedding horizon radius \(R_H = 1\).

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In the case of Kerr spacetime the Gaussian and mean curvatures of the embedding diagram are more detailed functions given by,

$$\begin{aligned} K = \frac{m \left( -a^6 (5 m+3 r)+a^4 r \left( 8 m^2+2 m r-7 r^2\right) +a^2 r^4 (11 m-5 r)-r^7\right) }{r^4 \left( a^2 (2 m+r)+r^3\right) ^2} \end{aligned}$$
(32)

and

$$\begin{aligned} H&= \frac{1}{2 F} \left( \frac{\Delta }{a^2 r^5 (2 m+r)+r^8}\right) ^{3/2}\bigg \{ \frac{r^{11/2} \left( G^2 \left( r^3-a^2 m\right) ^2-3 A P^4\right) }{\sqrt{\Delta } G^2} \nonumber&\\ -&\frac{A r^{13/2} (m-r) \left( 2 a^4 m^2-a^2 m r^3-r^6\right) \left( 6 a^4 G m^2+3 a^2 G m r^3\right) }{\Delta ^{3/2} G P^6}+\frac{A F^4 \sqrt{r}}{\Delta ^{3/2}} \nonumber&\\ +&\frac{A r^{13/2} (m-r) \left( a^2 m-r^3\right) \left( 2 a^4 m^2-a^2 m r^3-r^6\right) \sqrt{16 a^4 m^2+10 a^2 m r^3+r^6}}{\Delta ^{3/2} G P^5} \nonumber&\\ -&\frac{3 a^2 A m r^{11/2} \left( a^2 m-r^3\right) \left( 16 a^4 m^2+10 a^2 m r^3+r^6\right) ^{3/2}}{\sqrt{\Delta } G^3 P^5} \nonumber \\&+\frac{3 A r^{17/2} \left( -5 a^4 m^2+4 a^2 m r^3+r^6\right) }{\sqrt{\Delta } G^2 P^2} +F^2 \sqrt{\frac{r}{\Delta }} \left( r^3-a^2 m\right) ^2 \bigg \} \end{aligned}$$
(33)
$$\begin{aligned} A=&a^2 (2 m+r)+r^3 \; ; \; G=\sqrt{8 a^2 m+r^3} \; ; \; P=\sqrt{2 a^2 m+r^3} \end{aligned}$$
(34)
$$\begin{aligned} F=&\sqrt{\frac{m \left( -a^6 m+a^4 r \left( 2 m^2-m r+2 r^2\right) +4 a^2 r^4 (r-m)+2 r^7\right) }{a^2 (2 m+r)+r^3}} \end{aligned}$$
(35)

As a check point it is easy to see that for \(a=0\) the above curvatures reduce to their Schwarzschild values in (20).

Figure 8
figure 8

Behaviour of the embedded diagrams of different spacetimes, all starting from the horizon radius \(R_H =1\).

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Figure 9
figure 9

Behaviour of the embedding diagrams in the neighbourhood of their common horizon.

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Figure 10
figure 10

Behaviour of the Gaussian curvatures of all the embedding diagrams.

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Figure 11
figure 11

Behaviour of the Mean curvatures of all the embedding diagrams.

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Comparison of embedding diagrams and their curvatures

To compare the effect of different parameters on the embedding diagrams discussed above, and also compare their asymptotic behaviour, one can draw all the embedding curves in the same plot. To this end we plot all the curves starting from the same horizon size in terms of the embedding radial coordinate, namely \(R_H=1\). As noted previously, by choosing a given value for \(R_H\) the corresponding spacetime parameters are either fixed, as in the one-parameter solutions, or are restricted as in the case of two-parameter solutions. The behaviour of the embedding diagrams for all the above-considered spacetimes are given in Fig. 8.

Since the behavior of embedding diagrams in Schwarzschild, NUT (with \(m > l\)) and Kerr spacetimes for equal or almost equal masses look very similar, in Fig. 9 we have magnified those diagrams in the neighbourhood of their common horizon.

Gaussian and mean curvatures of all the embedding diagrams are also plotted against R, in Figs. 10 and 11 respectively. Obviously as expected, all the embedding diagrams are open 2-surfaces with negative Gaussian curvatures.

Dynamic embedding of NUT spacetime

Unlike spatial embedding diagrams discussed in the previous sections, which lack information on the spacetime dynamics, the spacetime or dynamic embedding of a radial plane ((t, r)-plane) into the three-dimensional Minkowski spacetime includes the dynamics of the original spacetime2. These diagrams include regions on both sides of a horizon. In the case of NUT spacetime (Eq. 4), the radial plane obtained by setting angular coordinates \(\theta\) and \(\phi\) equal to constants, is given by

$$\begin{aligned} ds^2 = f(r) dt^2 - \frac{dr^2}{f(r)} \end{aligned}$$
(36)
$$\begin{aligned} f(r)=\frac{r^2-l^2-2 m r}{r^2+l^2} . \end{aligned}$$
(37)

This is very similar to the radial plane of any static, spherically symmetric spacetime, including Schwarzschild and Reissner-Nordstrom black holes apart from a different function f(r) which now includes both the mass and the NUT parameters. Following2, we try to embed this radial plane into the three-dimensional Minkowski spacetime with the following metric in the Cartesian coordinates;

$$\begin{aligned} d s^2=d T^2-d X^2-d Y^2. \end{aligned}$$
(38)

To do so first we note that the above metric could be written in cylindrical hyperbolic coordinates (Rindler coordinates), i.e,

$$\begin{aligned} ds^2 = \rho ^2 d \zeta ^2-d \rho ^2-d Y^2 \end{aligned}$$
(39)

by the following coordinate transformations,

$$\begin{aligned} \zeta&= {\text {tanh}}^{-1}\left( \left( \frac{T}{X}\right) ^{ \pm 1}\right) \end{aligned}$$
(40)
$$\begin{aligned} \rho&= \sqrt{ \pm \left( T^2-X^2\right) }, \end{aligned}$$
(41)

where different combinations of ± signs give transformations for different Rindler wedges.

On the other hand the metric of the local geometry near any smooth non-degenerate horizon, including that of the radial NUT at \(r_H\), could be written in a \(1+1\) Rindler-like form. So the first step is to write the radial NUT metric in this form. To do so we use the definition of the surface gravity of the generic metric form (Eq. 36), i.e \(\kappa =\frac{1}{2} \partial _r f(r)|_{r=r_H}\), and the following coordinate transformations

$$\begin{aligned} \zeta&= \kappa t \end{aligned}$$
(42)
$$\begin{aligned} \rho&=\frac{\sqrt{f(r)}}{\kappa } \end{aligned}$$
(43)

which upon applying to Eq. (36) lead to the desired form

$$\begin{aligned} d s^2=\rho ^2 d \zeta ^2-d \rho ^2, \end{aligned}$$
(44)

near the horizon, by taking the leading term in the expansion of f(r) at \(r=r_H\). This is the \(Y=constant\) surface in metric (Eq. 39), and we need to find a general embedding surface \(Y = Y (\zeta ,\rho ) \equiv Y (t,r)\) to go beyond the above leading approximation. To this end we follow the same procedure employed in2 for the static spherically symmetric spacetimes. But before doing so we note that our ability to use the above procedure in Schwarzschild spacetime was reflected in the fact that one could use similar hyperbolic transformation as in Eq. (40), to maximally extend Schwarzschild metric, and obtain the Kruskal spacetime which divides the spacetime into four different regions. This is compared to Minkowski spacetime as the maximal extension of the Rindler geometry.

That we are allowed to use the above procedure in the case of NUT hole, is due to the fact that one can maximally extend the Taub-NUT metric with a kruskal-like extension35,36. This is discussed briefly in the appendix (Supplemental Material to this article). Now going back to the surface \(Y = Y (\zeta ,\rho )\), its equation can be obtained by equating the two metrics (36), and (39) at a given time \(t=\frac{\zeta }{\kappa }\), leading to

$$\begin{aligned} d \rho ^2 + dY^2 = \frac{dr^2}{f(r)}, \end{aligned}$$
(45)

which upon taking into account (Eq. 43), we get

$$\begin{aligned} Y=\int _{r_H}^r \sqrt{\frac{1}{f(r)}\left( 1-\frac{1}{4 \kappa ^2}\left( \frac{d f}{d r}\right) ^2\right) } d r. \end{aligned}$$
(46)

Now for the NUT spacetime with \(\kappa =\frac{1}{2 \left( \sqrt{l^2+m^2}+m\right) }\), the above equation together with Eqs. (42) and (43)comprise the following embedding equations for the NUT region \(r_H<r < \infty\),

$$\begin{aligned} \zeta&= \frac{t}{2 \left( \sqrt{l^2+m^2}+m\right) } \end{aligned}$$
(47)
$$\begin{aligned} \rho&= 2 (\sqrt{l^2+m^2}+m)\sqrt{\frac{r^2-l^2-2 m r}{r^2+l^2}} \end{aligned}$$
(48)
$$\begin{aligned} Y&= \int _{r_H}^r \sqrt{\frac{r^2+l^2}{r^2-2 m r-l^2}\left( 1-\frac{4 \left( \sqrt{l^2+m^2}+m\right) ^2 \left( l^2 m-2 l^2 r-m r^2\right) ^2}{\left( l^2+r^2\right) ^4}\right) } \, dr. \end{aligned}$$
(49)
Figure 12
figure 12

Dynamic embedding diagram for pure NUT spacetime with \(m=0\) and \(l=1\).

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Figure 13
figure 13

Dynamic embedding diagrams for the Pure NUT (\(l=1\)), and Schwarzschild (\(m=1/2\)) spacetimes.

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Figure 14
figure 14

Dynamic embedding diagrams for NUT spacetime with two different sets of values for its parameters.

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Embedding diagram of radial pure NUT spacetime (\(m=0\)) is shown in Fig. 12. The same embedding diagram for the two limiting cases of the NUT spacetime, namely pure NUT, and the Schwarzschild (\(l=0\)) spacetimes, with the same horizon at \(r_H = 1\), are illustrated together in Fig. 13. Embedding diagrams of NUT spacetimes with two different sets of values for the mass and NUT parameters are shown together in Fig. 14. In this case, to compare the effects of mass and NUT parameters, we have a NUT spacetimes with a small mass parameter, and another with a small NUT parameter.

As expected, the null lines \(X=\pm T\) (corresponding to \(r=r_H\), and equivalently to \(Y=0\)) in these diagrams divide the surface into four regions. Two of them, extended to positive values of Y, are related to the outside of the horizon (corresponding to the two \({\textrm NUT}_+\) regions), while those with negative values of Y are concerned with \(r < r_H\) ( corresponding to the two Taub regions). On these diagrams two different families of coordinate curves are drawn which are the \(r = constant\) and \(t = constant\) curves in terms of the original Schwarzschild-like coordinates in NUT spacetime metric (Eq. 4).

Since the NUT spacetime is asymptotically flat, the analysis of the information encoded in these diagrams is very similar to that of Kruskal spacetime, except the facts that the interior region is not a black hole region but a Taub region, and the \(r=0\) is not a singularity but a regular point.

Conclusions

We have obtained both the spatial and dynamic embedding diagrams for some stationary spacetimes including NUT and pure NUT spacetimes, and discussed their characteristics in comparison with the corresponding embedding diagrams in Schwarzschild spacetime. In the case of spatial embedding diagrams by comparing them and their Gaussian and mean curvatures, one can summarize the results, and the effect of different parameters as follows:

  1. (1)

    The expected spatial asymptotic flatness of all the spacetimes considered is obvious both from Figs. 9 and 10. Specifically, in Fig. 10 they all reach zero Gaussian curvature asymptotically.

  2. (2)

    From Fig. 9 it is obvious that in the NUT-type spacetimes (those having only mass and/or NUT charge), having larger mass (smaller NUT parameter) leads to a more steep diagram. In other words at a given distance, say \(R=1.4\), the spacetime with a larger NUT parameter (smaller mass) has a more negative Gaussian curvature.

  3. (3)

    Figures 9 and 10 show that both Gaussian and mean curvatures of NUT-type spacetimes start from the same curvature value at the horizon, while those of R–N and Kerr spacetimes start from higher different curvature values. One may take this observation as an evidence against interpreting the NUT parameter l as a measure of a kind of an angular momentum as Bonnor did16. On the other hand one may find it more consistent with its interpretation as a twist parameter25.

  4. (4)

    Finally it is noted from Fig. 10, that spacetimes with large NUT parameters (\(l=0.65\) and \(l=\sqrt{0.5}\)), have a different behaviour in the change of their Gaussian curvatures near the horizon as compared to those either with a small NUT parameter or without it.

One should be very careful with drawing conclusions from either of the spatial or dynamic embedding diagrams about the intrinsic properties of the corresponding spacetimes and their effects on particle trajectories, specially those related to spacetime curvature. For example, the geometry of the Flamm’s paraboloid could help us to calculate the contribution of the spatial geometry on the precession of a planet’s perihelion, as well as on the light bending29, but not the full spacetime contribution. In the case of NUT spacetime, it is already known that a small NUT parameter will induce a small precession in the timelike orbits5. This result seems to be in agreement with the small change in the spatial embedding diagram from that of Schwarzschild by adding a small NUT parameter. This small difference is more distinguishable at larger values of R as could be seen in Fig 8 for the embedding diagram for \(l=0.12\).

On the other hand, in the case of dynamic embeddings, the inclusion of the time coordinate provides these diagrams with a dynamic nature so that they include more visually traceable features of particle trajectories in the corresponding spacetime. From the relative behaviour of these trajectories, embedded in \((1+2)\)-dimensional Minkowski spacetime, one could obtain physical effects rooted in the curved nature of the underlying spacetime. These include gravitational redshift and gravitational tidal effects near the horizon and far away from it in the asymptotically flat region of the spacetime2.