# Schwarzschild Spacetime And Black Holes

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Abstract. We introduce and discuss the Schwarzschild metric, at a level suitable for beginners with basic calculus knowledge. First, the exterior vacuum metric is examined, and several techniques about how to work with metrics are demonstrated, and their physical significance explained. A geometric understanding of the various terms in the metric is developed, and some fully worked examples are given. We then progress on to the interior metric, and present its peculiarities and physical consequences. This leads on to a discussion of Schwarzschild black holes, the meaning of event horizons, and what physically happens if we allow a test particle to freely fall into such black holes, from the points of view of different observers. Animations and a video are presented that help visualise these principles.

This article deals with the simplest non-trivial solution to Einstein’s field equations, which was also the first closed analytical solution that was found – Schwarzschild spacetime. This solution is based on the following boundary conditions :

1. The body is isolated in an otherwise completely empty space; there are no external influences, gravitational or otherwise, and no other gravitational sources anywhere
2. Spacetime becomes asymptotically flat at infinity, i.e. the laws of gravity reduce to Newton’s inverse square law at great distances from the central body
3. The body is spherically symmetric
4. The body is static and stationary ( i.e. it doesn’t move, and its gravity does not change over time )
5. The body does not carry angular momentum ( no rotation )
6. The body carries no net electric charge

So basically, we are examining the geometry of spacetime inside and outside a spherical body that doesn’t move, isn’t charged, doesn’t rotate, and is not affected by any other outside influences. It should be immediately obvious that this scenario is an idealization – no real world body will ever strictly fulfill all of these conditions. Even the simple presence of a test particle which moves in Schwarzschild spacetime strictly speaking already precludes the very existence of this geometry, since the central body ceases to be isolated in an otherwise empty space ( condition (1) ). Nonetheless, as it turns out, the Schwarzschild solution proves to be an excellent model for the gravitational influence of many objects in the real universe, not least of which being the planets in our solar system, and the Sun itself. Note that for now we do not make any assumptions as to the nature of the central body – the Schwarzschild metric describes spacetime outside “ordinary” bodies such as planets and stars just as well as it does describe the exterior of more exotic phenomena such as black holes, so long as all of the above conditions hold.

We will start by taking a closer look at the exterior Schwarzschild metric first, which is a description of spacetime in the exterior vacuum outside the central body; the second part of this article then briefly deals with the interior Schwarzschild metric, being a description of spacetime inside the central body. Finally, we take a closer look at Schwarzschild black holes, and their associated physics.

#### Exterior ( Vacuum ) Schwarzschild Spacetime

Recall the mathematical form of the exterior solution :

(1)

Herein – and for much of the rest of this article – I am following Taylor/Wheeler’s convention of using geometrized units, where everything is expressed in units of meters [1]. The parameter M that appears in the above metric is the mass of the central body expressed in units of meters, which is related to units of kilograms as per

(2)

The differential is the surface element of a unit sphere :

(3)

This simple fact – the latter part of the metric being that of a unit sphere – provides a crucially important insight as to the geometric structure of Schwarzschild space-time : if we hold the time coordinate t and the radial coordinate r constant, we end up with a plain old sphere. That means that Schwarzschild spacetime is quite simply a collection of nested spheres, each one of which corresponds to a choice of radial coordinate r, and is subject to the normal, well known relations of spherical geometry [2] :

Schwarzschild Spacetime. The geometric structure of Schwarzschild spacetime is that of a family of nested spheres.

Each one of these spheres ( i.e. surfaces of constant t and constant r ) taken on its own is somewhat trivial – it is really just a simple, normal sphere, described by the standard line element (3); there are no deviations from standard spherical geometry. This can be seen in the Schwarzschild metric (2) by the fact that there are no coefficients in front of the angular part, other than the radius r itself.

In order to understand Schwarzschild spacetime, it is furthermore important to develop a geometric understanding of the metric (2), or else one is at risk of falling into one of the many pitfalls and misconceptions that often go hand in hand with this solution. We have already taken the first step by realising that Schwarzschild spacetime is really just a collection of nested spheres, and each sphere on its own is not “distorted” in any way, i.e. it is subject to the normal laws of spherical geometry. In particular, if we hold t constant and choose a particular radius , then the surface area of the sphere corresponding to this is

(4)

and its Gaussian curvature is

(5)

This provides the physical meaning of our radial coordinate r – it is chosen such that each r=const. corresponds to a standard sphere.

Schwarzschild Coordinates. The coordinates and have the same meaning as for spherical coordinates in flat space. However, r cannot be interpreted as radial distance measured by any ruler, and t cannot be interpreted as an interval of time measured by any clock; these coordinates are just accounting devices, and are not measurements taken by any real-world observer. Actual physical measurements that correspond to real observers must be computed from the metric.

Let us hold all other coordinates constant, and just vary r only; the metric (2) then becomes

(6)

To understand what this means, let us plot this relationship graphically [3] :

If we decrease the coordinate separation dr by one unit ( i.e. we go closer to the central body ), we increase the proper distance ( the bold line on the bottom ) between adjacent spheres, as given by the relationship (6) from the metric. This means that, the closer we get to our central body, the more “spaced-out” our family of nested spheres becomes – space is “stretched” in the radial direction near a massive body. At the same time, the circumference of the circles in the above diagrams also reduces; however, if we vary the radius by one unit, the circumference does not in fact reduce by as expected, but by a lesser amount [4] :

This is again a manifestation of the relationship between spheres not being trivial in this geometry – both circumference and distance vary in non-trivial ways, if we vary the r coordinate. The variation itself is a function of both the mass of the central body M, as well as the radial distance from it.

It is instructive to discuss the physical significance of the coordinates used in the metric (2) as well. Imagine you are an astronaut, and you are stationary somewhere very far away from the central body ( “at infinity” ), and you are looking out the window of your rocket; in many ways, this is like a “God’s eye” view of the spacetime around the gravitating mass. You now take a marker pen, and start drawing concentric circles on the glass of your window, with the central body being at the centre ( the locus ) :

Each circle is spaced one unit from the next, and each circle represents a sphere with surface area . What you arrive at is a way to label nested round spheres as seen by a stationary observer at infinity ( i.e. in flat spacetime ). The radius r is chosen such that each sphere has surface area , meaning r is chosen so as to preserve spherical geometry as seen from infinity, but it is not an accurate reflection of proper ( =physical ) radial distances between spheres for those who actually travel there ! The relationship between how an observer at infinity labels the spacing between spheres ( dr ), and what the local geometric distance between the same two spheres ( ) actually works out to, is as depicted in fig 2 above.

Schwarzschild Radial Coordinate. The coordinate differential dr in the Schwarzschild metric denotes how radial coordinate separations between nested spheres change for a stationary observer at infinity; the differential  denotes how proper geometric distances between spheres change locally ( not at infinity ) between them. The relationship between the two is given by (6). The radial coordinate is defined such that each nested sphere as surface area (4) and Gaussian curvature (5), meaning that it is not an accurate representation of ( physical ) radial distance, but rather simply an accounting device.

Coordinate vs Proper Quantities. Proper quantities are those that are physically measured in a local frame, using clocks, rulers and accelerometers. Coordinate quantities are those that are calculated by distant observers using their own method of labelling events. In curved spacetime, the two do not generally coincide, and must be carefully distinguished. All observers agree on proper quantities, while coordinate quantities are strictly observer-dependent.

What we have here is an example of the principle of locality – all measurements of space and time are purely local, and in general valid only for whichever observer performs them using his own clocks and rulers. Coordinate measurements are those performed far away, whereas proper measurements are those performed locally; in curved spacetimes, the two are related in non-trivial ways, as given by the metric.

If we bring all we have learned so far together, we can combine it into a so-called embedding diagram of Schwarzschild spacetime [5] :

The bottom ( flat circle ) is what a far-away stationary observer calculates using his own local ( i.e. far-away ) method of labelling events, whereas the top part ( the cone ) is what an observer who physically resides somewhere in Schwarzschild spacetime will measure and experience, again using his own local method of labelling events. These two coincide only when at rest at infinity. As you can see, everywhere else, there is quite a discrepancy between the two – what someone calculates and measures locally far-away ( dr – radial coordinate distance ) is not the same as what somewhat measures and experiences locally in the vicinity of the central body ( ds – proper distance ); that is precisely the difference between coordinate and proper quantities, and it is precisely what it means for a spacetime to be curved. Proper quantities and coordinate quantities no longer coincide across extended regions.

Take careful note that this type of embedding diagram represents a “slice” of constant time, i.e. it depicts only spatial distances, but no time-like separations. They are thus not to be confused with the length of the world lines of physical test particles !

To get a real “feel” for what is physically happening here, let us work through a specific, numerical example [6]. Suppose we have a central body ( let us consider it as a point mass ) of one solar mass, and we want to know how far apart the spheres r=4km and r=5km are. The coordinate distance between the two is quite simply and trivially 5km – 4km = 1km, and this is what a far-away stationary observer would calculate when he looks out the window of his rocket ship and performs his own measurements with his own local ( =far away ) rulers. But what is the proper distance between the two spheres, as physically measured by someone who is physically there ? To answer this, we need to start with relation (6), which provides the relationship between increments in coordinates, and increments in physical measurements. Because the coefficient is a function of the coordinate r, and therefore varies between r=4km and r=5km, we need to perform an integration :

(7)

This isn’t a common or especially simple integral to evaluate, but if we perform the substitution , we can arrive at something we can work with :

(8)

Inserting all the numbers, we get a final result of . What this means is that, while the distance between the spheres r=4km and r=5km in this example is just 1km from the point of view of someone stationary very far away, it is 1723km for someone who physically and locally measures the distance between these two radial points. This goes to show just how divergent the points of view of different observers in Schwarzschild space-time can be, and how much it depends on what method of labelling events is being employed. It is therefore of crucial importance to never confuse coordinate quantities with proper quantities.

This being said, Schwarzschild coordinate quantities are not just mathematical oddities, but they have physical reality – they represent the reality of someone who watches a region of Schwarzschild spacetime while remaining stationary and far away from it. They are a valid description of local reality for a stationary observer at infinity, but as such they are not to be confused with a valid description of what happens locally close to the central body itself.

Everything we have said this far applies to time as much as it does to radial distance, with the only difference being that the temporal relationship is the inverse of the spatial one, in that time dilates the closer you get to the central body. When coordinate time – which is again what a stationary observer at infinity calculates and measures with his own clock – is increased by dt, the proper time which an observer at radial distance r physically records on his clock increases by

(9)

which is less, meaning an outside observer calculates a clock located closer to the central body to tick “slower”. This is known as gravitational time dilation, and is a relationship between clocks at different places in a gravitational field.

Schwarzschild Metric. In Schwarzschild spacetime, proper lengths are more and more stretched and proper times are more and more dilated the closer you get to the central body, compared to reference measurements taken somewhere far away.

Those of you who are especially observant will have noticed something odd – the metric coefficient for the radial coordinate diverges for , and becomes zero in that limit for the time coordinate [7] :

The radial point is called the Schwarzschild radius. As the Schwarzschild radius is approached, the ratio between coordinate distance and proper distance grows without bound, and the ratio between coordinate time and proper time asymptotically approaches zero. The Schwarzschild coordinate system has a singularity at this point, and it also has an obvious singularity at ; the metric is not defined at either one of these regions. However, these two singularities are physically very different; to see why, we take a closer look at the Riemann curvature tensor, or more specifically, at an invariant quantity formed from it called the Kretschmann scalar. This scalar is obtained by contracting all four indices of the curvature tensor, and, for the case of the Schwarzschild solution, evaluates to

(10)

It is not so easy to give this scalar quantity an intuitive interpretation, but you can think of it as a measure of the total strength of all tidal forces in the various directions at a given point; as such it is much like an average measure of curvature around a given point. Its significance as a tool in GR is that it allows us to investigate the nature of metric singularities such as at and . For the case of , the Kretschmann scalar diverges just as the metric does, so spacetime in this region is clearly not smooth and regular by any definition – this type of singularity is called a curvature singularity, and it indicates a region where curvature ( and hence geodesic deviation ) grows without bound. On the other hand, the region where  yields a Kretschmann scalar that is finite and well defined, indicating that spacetime here is smooth and regular. This type of singularity is called a coordinate singularity, and is merely an artefact of our choice of coordinates, much like spherical coordinates become singular at the poles. Coordinate singularities can be eliminated by a simple coordinate transformation, whereas curvature singularities can not.

Singularities. Coordinate singularities are artefacts of a particular choice of coordinates, and do not imply any irregularities in the structure of spacetime itself. Curvature singularities on the other hand are regions where spacetime itself becomes irregular, and curvature grows without bound there. This is a physical phenomenon and cannot be eliminated by going into a different coordinate system. One can distinguish between these by examining the Kretschmann scalar, which is formed from the Riemann tensor.

For the majority of real, physical bodies, the above has little significance since the region across which their mass-energy is distributed is very much larger than the corresponding Schwarzschild radius ( planets, stars, etc etc ), so this hypothetical “surface” is never exposed in vacuum. However, there are objects the entire mass-energy of which is contained inside a region smaller than the associated Schwarzschild radius – such objects are called black holes, and the hypersurface defined by their Schwarzschild radius is called an event horizon, for reasons that will become clear shortly.

Black Holes. An object, the entire mass-energy of which is located inside its Schwarzschild radius, is called a black hole. For such objects, the Schwarzschild radius is “exposed” in the exterior vacuum, and is called an event horizon.

We will look at black holes in the third part of this article ( see below ). For now, let us have a closer look at the dynamics of test particles in Schwarzschild spacetime outside ordinary bodies; all of this will apply to black holes as well, with the only difference being that there are no event horizons for ordinary bodies. Before we can do this, we need to be clear about the various types of observers one can have in Schwarzschild spacetime.

Schwarzschild observers are purely hypothetical observers who reside at rest infinitely far away from the gravitational source ( i.e. in asymptotically flat spacetime ). They have – in a manner of speaking – a “global” point of view in the sense that they can be considered “bookkeepers” for the way other observers label events in their own coordinate systems. Schwarzschild coordinates are therefore also called bookkeeper coordinates; no physical observer corresponds to these coordinates, they are merely an abstract accounting system that combines measurements taken by more local shell and free-fall observers ( see below ), and translates them into labels within the entire spacetime. We must therefore be very careful how we physically interpret relations expressed in bookkeeper coordinates, since these coordinates do not accurately measure radial distance or separations in time. We will denote Schwarzschild coordinates by ordinary Latin letters, as in for example dt.

Bookkeeper Coordinates. Schwarzschild coordinates are also called bookkeeper coordinates, and they represent a global accounting system to keep track of local measurements across the global spacetime. They do not correspond to any physical observer.

The second type of observer we encounter is the shell observer. This type of observer resides stationary at a fixed, finite radial distance ( i.e. r=const. ), for example a rocket ship firing its thrusters such that it remains stationary with respect to the central body. The main difference to Schwarzschild coordinates is that the distance to the central body is finite and well defined, hence the name “shell” observer. We will denote shell coordinates with the appropriate subscript, for example .

Shell Observers are observers which remain stationary at a finite, constant radial distance from the central body.

The last type then is the free fall observer – as the name implies, this is an observer who is allowed to fall freely under the gravitational influence of the central body. If that free fall starts from rest at infinity, the associated coordinate system is called rain coordinates, or Gullstrand-Painlevé coordinates. If the free fall starts at infinity, but with some initial velocity v, the coordinate system is called hail coordinates; if it starts from rest at a finite distance, they are called drip coordinates. These coordinates are denoted again by appropriate subscripts, such as  or .

Free Fall Observers are observers which are allowed to fall freely under the influence of gravity, subject to initial conditions.

To understand dynamics of test particles, we need to recognise the fact that there are certain constants of motion in Schwarzschild spacetime; these are quantities that remain conserved everywhere along a test particle’s world line, and they arise from symmetries of the metric. Firstly, the metric coefficients in (2) do not depend on time t, which yields a constant of motion called energy per unit rest mass :

(11)

This provides a relationship between energy, coordinate time, and proper time; the energy E for a particle in free fall remains conserved for all values of r. Note that, unlike in Newtonian physics, the quantity E cannot be uniquely split into a kinetic and a potential part. For those of you who are interested, the reason for that is that the underlying Lagrangian of Schwarzschild motion is nonlinear.

Secondly, we find that the coefficients of the Schwarzschild metric also do not depend on either of the two spatial angles, this leads to a second conserved quantity called angular momentum per unit rest mass :

(12)

Constants of Motion. In Schwarzschild spacetime, energy per unit rest mass and angular momentum per unit rest mass are conserved at every point along a free fall trajectory.

From these simple definitions, we can already deduce a number of important relations by simply forming the ratios between various differentials from the metric. For a Schwarzschild observer at infinity, the radial coordinate velocity ( i.e. the velocity taken along the radial direction only ) for a freely falling test particle is

(13)

Note that this is a coordinate quantity – it reflects only what a very distant observer calculates based on his own local measurements, but it does not reflect what a local shell observer hovering near the central body actually sees. A shell observer will instead measure the radial velocity

(14)

The difference between these is important, especially when we approach the Schwarzschild radius – Schwarzschild coordinate velocity ( seen from very far away ) goes to zero here, whereas local shell observer coordinate velocity ( seen from someone hovering just above the Schwarzschild radius ) goes towards the speed of light c. Similar disagreements also happen with energy; the energy detected by a local shell observer, and the energy calculated by a distant Schwarzschild observer, are related via

(15)

Note that while E ( Schwarzschild energy per unit rest mass ) is a constant of motion, is not – it is a purely local quantity which depends on the position of the shell observer.

It is possible – with a bit of effort – to form all and any relationships along these lines; it is neither possible nor desirable in the context of this article to list them all. I refer the interested reader to the excellent text “Exploring Black Holes” by Edwin Taylor and John Archibald Wheeler for a more thorough treatment. I will only briefly mention one more important quantity, which is the proper time a clock physically records while freely falling from rest at infinity ( rain frame ), between two given radial points :

(16)

All of these quantities describe aspects of the same world line, just as the various observers listed above describe the same Schwarzschild spacetime, albeit from different perspectives, and using different methods to label events. Formally, the metrics of the various observers are related via coordinate transformations, so they are just different expressions for the same spacetime geometry. We are at liberty to choose the point of view from which a particular problem at hand can be most easily examined.

Spacetime and Coordinates. The Schwarzschild metric can be expressed in different coordinate systems, which physically corresponds to different observers in different states if motion. While the mathematical form of the metric may look very different between different choices of coordinate basis, they all describe the same spacetime with the same geometry, just from different points of view. All of these metrics are related via coordinate transformations, and are therefore physically equivalent ( principle of covariance ).

So far, everything that happens above in figures 2, 3, 4 and 5, and the associated explanations, is all taking place on the same “slice” of constant time – this means, we were purely looking at the geometric distance in space only between the nested Schwarzschild spheres. Likewise, in (9) we were looking at distances in time only between events. However, real-world test particles travel neither in time nor in space alone – their world lines represent separations between events in spacetime. As can be seen in the Schwarzschild metric (1), both the radial and the time coordinate change in non-trivial ways, and they both explicitly depend on the r coordinate; to examine the form and geometry of actual world lines, we therefore need to find an equation of motion that can accurately account for all of these changes as a test particle freely falls under the influence of gravity. The most general form of this equation of motion is precisely the geodesic equation, which is the mathematical version of the physical statement that accelerometers in free fall read exactly zero at all times :

(17)

or, with the covariant derivative explicitly written out :

(18)

This differential equation is valid for any spacetime ( i.e. for any given metric ), and describes the set of all possible geodesics therein; using our boundary conditions 1-6 from the beginning of this article, we can obtain the specific solution for the Schwarzschild metric, which is most often written in terms of the constants of motion, as well as a function called the effective potential . The full and formal solution of (18) is the world line of a test particle through spacetime, given a set of initial conditions ( initial position and velocity ); for the purpose of this article however, we are mostly interested in the radial motion of test particles, meaning we restrict our attention to how the radial coordinate of the world line changes with respect to our chosen time coordinate. If we are able to parametrise the world line using proper time ( i.e. for massive test particles ), we obtain

(19)

with the effective potential function

(20)

The equation of motion (19) is a formal solution of the full geodesic equation (18) for the metric (1), as one can readily verify – the actual process of solving the differential equation is lengthy and quite tedious, so I will skip it here ( see any textbook on GR for details ).

If we plot this effective potential, we can quickly and at a glance see what orbits are possible in Schwarzschild spacetime ( the following is for a particular value of L/m = 4M ), in terms of the constants of motion [8] :

To compute the full orbit of a test particle, in terms of both radial and azimuthal motion, one can proceed as given in the following general prescription taken from [9] :

To obtain a similar equation of motion for massless particles ( i.e. photons ), we need to choose a different affine parameter to parametrise our world line, since proper time is not defined for photons. It is often convenient to forgo the difficulties associated with choosing a general, non-physical affine parameter, and instead consider the trajectory of light in terms of bookkeeper coordinates and a quantity called the impact parameter [10] :

The equations of motion for light in Schwarzschild spacetime then become

(21)

and

(22)

This once again allows us to calculate the full trajectories for rays of light, in terms of radial and azimuthal motion in Schwarzschild spacetime [16] :

In order to get a more interactive feel for the possible results, I recommend you play around with this Wolfram Demonstrations applet – this works for both massive ( time-like ) and massless ( null ) test particle geodesics; simply change the parameters as required, and watch what happens.

Some specific consequences of motion in Schwarzschild spacetime deserve explicit mention. First of all, light, even though photons are massless, are subject to the same laws of gravity as any other test body; that means that a ray of light will experience deflection around a massive body [11] :

This deflection of light is due to two factors : the curvature of space itself, as well as the fact that light is in free fall under the influence of gravity, just like any other test body. The total deflection angle is therefore the result of both straight lines failing to remain “straight” with respect to a distant reference line, as well as radial motion due to free fall under gravity [12] :

It is possible to also model light deflection using the old Newtonian model of gravity – however, because Newtonian space is always flat and Euclidean, the failure of lines to remain parallel due to spatial curvature isn’t accounted for, so the deflection angle in GR is twice what it would be in Newtonian gravity. This result has been observationally tested and verified many times.

Another peculiarity of Schwarzschild orbits is a phenomenon known as perihelion precession. In Newtonian celestial mechanics, the frequency at which a body completes its orbits, and the frequency at which it oscillates radially while doing so, is necessarily equal, since the notion of “time” is global and hence the same everywhere [13] :

Not so in General Relativity – here, time becomes a local notion, so the temporal relationships between events at different points within the orbit are non-trivial. This leads to a small difference between angular and radial frequency, so that the perihelion of the orbit advances slightly with each completion [14] :

This is again a phenomenon that has been observationally verified, most notably for the planet Mercury, where it is quite apparent due to its proximity to the sun.

Lastly, the curvature of spacetime around a massive body leads to an effect called the Shapiro delay. If you send a ray of light ( or a radar beam ) from an emitter to a detector such that is makes a close pass at a massive body, you will find that the arrival time of the beam at the detector is delayed, compared to what it should have been based on the naive reckoning of an observer stationary with the receiver. Take for example the signal received from a pulsar orbiting a white dwarf [15] :

For a far-away observer ( e.g. Earth ), radially distances are equally separated throughout all of spacetime ( figure 4 above ), so he concludes that the beam of radiation must have slowed down as it passed the massive body, in order to account for the delay in arrival times. While this is certainly true in the distant observer’s local frame of reference, one must remember that this speed of light is a coordinate quantity – it is not directly measured at all, but only calculated based upon his own way of reckoning distances and measuring time, from a far-away point of view. Clearly, even though this way of explaining the effect is a valid conclusion in his own distant frame, it is a violation of a fundamental principle of relativistic physics, the principle of general covariance – the laws of physics must be the same regardless of where you are and how you move. The speed of light therefore cannot vary, since Maxwell’s equations are indeed generally covariant. So how is the Shapiro delay to be explained ? The answer is of course in figure 2 above – due to curvature of spacetime, the far-away observer’s method of calculating radial distances is not an accurate reflection of actual physical distances closer to the massive body; as seen in the above picture ( fig. 13 ), the trajectory of the beam of radiation is longer when it passes close to the central mass, as compared to the same situation further away. Therefore, the beam of radiation has to travel further and for longer, as measured by a far away clock. Locally at every point of the trajectory though, the speed of light is exactly c, always and everywhere.The Shapiro delay is, in this sense, a direct measurement of the curvature of spacetime itself, just as the Pound-Rebka experiment is. Just as with all other quantities in curved spacetimes, it is thus crucially important to not confuse the coordinate speed of light with the local proper speed of light; the former may vary from observer to observer, whereas the latter is always exactly c. Even at infinity, the two will not always coincide, as the Shapiro delay demonstrates.

Speed of Light. The proper speed of light – which is what is physically measured locally – is always exactly c everywhere, both in flat and in curved spacetime; this is a manifestation of the principle of covariance, which demands that Maxwell’s laws ( from which the speed of light directly follows ) are the same for all observers, once formulated appropriately. On the other hand, the coordinate speed of light – which a distant observer calculates, but never directly measures – varies between observers, and is not constant. The latter is valid only for far-away observers, and hence does not imply any changes in local physics, such as amendments to the laws of electrodynamics.

Let us briefly summarise the main points of what we have learned up until now. The exterior Schwarzschild metric is of the general form

(23)

with coordinate functions f(r) and g(r), which depend on the radial coordinate, but not on time. The spatial distance between nested spheres is

(24)

The proper time of a static observer between events on the same sphere is

(25)

The separation between events for a body in radial free fall from rest is

(26)

Schwarzschild spacetime is hence a family of nested spheres, each one of which is subject to the normal laws of spherical geometry.

Thus far, everything we have discussed was valid strictly for the vacuum outside a massive body only. Before we go any further, we need to briefly discuss what the geometry of spacetime is like in the interior of a mass distribution.

#### Interior Schwarzschild Spacetime

The interior geometry of spacetime, i.e. the geometry inside a distribution of energy-momentum ( that can be either mass-energy, or fields ) is very much more complicated than their exterior counterparts, but conceptually, the same principles and concepts apply. We start off by using a metric ansatz that looks slightly different from the one used for the exterior solution, but encapsulates the same physics :

(27)

The two unknown functions and are known as source potentials. In order to find these, we demand essentially the same boundary conditions as for the exterior solution, with the following differences :

1. The interior solution must smoothly reduce to the exterior vacuum solution at the surface of the massive body ( i.e. no physical singularities or discontinuities must occur there )
2. The potential functions must reduce to zero at infinity ( Minkowski spacetime )
3. The mass-energy distribution can be approximately described as an ideal fluid

In order to derive the interior metric, we thus write down the energy-momentum tensor of an ideal fluid :

(28)

Herein, is the isotropic pressure in the fluid’s rest frame, p(r) is the density of mass-energy in the fluid’s rest frame, and is the fluid’s 4-velocity. The metric then is a solution to the interior Einstein equations

(29)

The exact analytical form of the solution very much depends on initial and boundary conditions, but a general prescription can be given as follows. The line element will be of the form

(30)

which contains the mass function

(31)

that describes the total amount of mass-energy enclosed by a given radius r. The pressure function p(r) is a solution to the Oppenheimer-Volkoff equation

(32)

The second source potential, , is a solution to the source equation

(33)

with the boundary condition ( R = surface radius of massive body )

(34)

which is necessary to ensure that the interior metric transitions smoothly into the exterior vacuum solution at the surface. Based on this, in order to find an explicit solution, the following steps are necessary :

1. Specify and equation of state for the ideal fluid
2. Specify a central pressure at r=0
3. Decide a value for the potential function at r=0
4. Renormalise p(r) so that p(r) = 0 is the body’s surface
5. Renormalise  so that all boundary conditions are satisfied

For most realistic equations of state ( that accurately describe real stars, planets, etc etc ), the resulting equations are too complicated to be solvable in closed analytical form, and can only be treated numerically. However, a closed solution is possible if we demand the fluid to be of uniform density, i.e.

(35)

for all values of r. The analytical expression is still lengthy and unwieldy, and adds little to an understanding of the geometry, so I will skip this ( see any textbook on the subject for details ) and present the corresponding embedding diagram [17] :

The shaded part on the bottom ( the “cusp”, the shading of which unfortunately isn’t clearly visible in this graphic ), represents the interior part of the metric, whereas the top, unshaded part is the exterior metric in vacuum. The two parts are smoothly joined, and everywhere regular. This metric therefore spans the entire spacetime, both interior and exterior of the gravitating body. The Schwarzschild radius is located inside the body, but it is not an event horizon, since it is not exposed – not all of the mass is inside this radius. The object is not a black hole, and the Schwarzschild radius has no special significance here.

It is important to realise the physical meaning of the interior part of the metric. It describes a distribution of mass-energy ( a star, planet, etc ) that is in an equilibrium state – the gravity that the body itself “generates” is exactly counterbalanced by the internal pressure of the material, leading to a stable configuration. The central pressure within a body of this type with mass M and radius R is

(36)

This is the maximum pressure available to act against gravity, thereby supporting the mass. If we hold the radius constant and add more mass, then more pressure is required to support the object; likewise, if we hold the mass constant and reduce the radius, then again the pressure will have to increase to support the equilibrium condition. At the same time, pressure is itself a source of gravity, leading to a highly non-linear situation. If we hold R constant and treat (36) as a function of mass, we find that the central pressure increases monotonically and without bound if we approach the limits [18] :

(37)

and/or

(38)

Physically this means that, if you have a body of the type discussed here, and you either increase its mass while holding the radius constant, or you hold mass constant and reduce the radius, then the pressure required to support this body against its own gravity increases without bound as you approach the limits (37) and (38). This is a purely relativistic phenomenon, and not predicted by Newtonian gravity – the laws of gravity that govern the interior of very dense and/or massive objects are radically different in terms of behaviour and dynamics from their Newtonian counterparts.

On the road to that upper limit, we find physical objects comprised of matter at various levels of degeneracy. In the first instance, we have ordinary bodies made up of atoms; the pressure required to support such objects against their own gravity comes from electromagnetic interactions between atoms, and the laws of quantum physics which describe the bonds between them. Such objects are often represented as white dwarf stars, and they remain stable so long as they stay within a limit called the Chandrasekhar limit. Once this limit is exceeded, electromagnetism no longer provides sufficient pressure to act against gravity, and further collapse takes place; individual atoms are destroyed, and ordinary matter is transformed into neutronium – this is a state of matter that can be characterised as a relativistic, dense and heavy degenerate neutron gas. The resulting object is called a neutron star, and the pressure that keeps it stable is a quantum mechanical phenomenon called neutron degeneracy pressure. There is again an upper limit, past which the degeneracy pressure becomes insufficient to prevent further collapse – this is called the Tolmann-Oppenheimer-Volkoff limit. If it is exceeded, the body will theoretically collapse further into a state called quark-degenerate matter, which is a gas made up of ( nearly free ) quarks and gluons, that generates its own type of degeneracy pressure. Such hypothetical objects are called quark stars – however, there is to date no direct observational evidence for the existence of such objects, and the properties of quark-degenerate matter are poorly understood due to the difficulty of modelling interactions involving the strong force. Nonetheless, if they exist then there will be an upper limit to the associated degeneracy pressure as well – and if that limit is exceeded, then there remains no known mechanism by which a further gravitational collapse could be halted, not even in principle. In essence, the body enters a process whereby its own gravity overwhelms all other known countereffects, be they classical or quantum mechanical.

The body will undergo a complete gravitational collapse, into a state where all its mass is concentrated in an infinitesimally small region – a state which is called a gravitational singularity. All curvature tensors diverge here, and all free fall geodesics terminate here, and cannot be extended further – this is known as geodesic incompleteness, and provides the formal definition for a singularity. The Schwarzschild radius is now exposed, in the sense that all mass-energy is confined within it, and called an event horizon. Such an object – a region of geodesic incompleteness behind an event horizon – is called a black hole.

Black Holes. As the ratio between mass and radius approaches the critical limits (37) and (38), the matter that comprises a massive body undergoes varies levels of degeneracy. The highest such level is quark-degenerate matter; if its associated degeneracy pressure is exceeded, there is no known mechanism in nature that can stop a complete gravitational collapse of the body into a singularity. The resulting object is called a black hole. Schwarzschild singularities are always shielded by event horizons, and are characterised by the property of geodesic incompleteness – all free fall geodesics terminate there, and cannot be extended either in space or in time.

At least, that is the conclusion arrived at by considering only General Relativity – the big caveat here is of course that GR is a purely classical theory, and does not in any way, shape or form account for quantum effects. As such, the formation of a singularity is not surprising – such entities are not to be understood as real, physical objects of zero volume and infinite density ( which is a nonsensical notion ), but rather as a sign that the theory has been wrongly extended past its domain of applicability, and hence breaks down and ceases to make sensible predictions. While there is little doubt as to the veracity of the overall concept – the “battle” between an object’s gravity and the pressures that counteract it – the final stages of a complete collapse are not something that General Relativity can adequately describe. To do this, we would need a full theory of quantum gravity, which, at the time of writing this article, does not yet exist.

Singularities. Gravitational singularities are not real physical objects predicted to exist by General Relativity, but rather to be understood as a sign that the theory ceases to be applicable in that region, and hence ceases to make sensible predictions there. This is because GR is a purely classical model, and does not account for quantum effects, which arguably become crucially important during the end stages of a gravitational collapse.

While GR is unable to correctly model the central singularity ( which in all likelihood won’t exist in any physical sense ), it is nonetheless very well able to describe the resulting black holes as macroscopic, astrophysical objects. We will therefore now turn our attention back to Schwarzschild spacetime in vacuum, and take a closer look at black holes.

#### Schwarzschild Black Holes

In order to fully understand the physics of black holes and event horizons, it is helpful to think about spacetime in terms of its causal structure. This means that we do not just consider it as a collection of events, but also attach a light cone to each of these events, which indicates to us which future physical trajectories are possible for a particle travelling through this event. Remember that light cones are a representation of causal structure, in that they depict which events can be causally connected and which can’t [19] :

The effect of spacetime curvature ( i.e. gravity ) is that it tilts light cones; in the case of Schwarzschild spacetime the tilt will always be towards the central mass [20] :

Gravity and Light Cones. In Schwarzschild spacetime, the effect of gravity is to tilt light cones towards the gravitating mass.

So long as we are in a region outside the Schwarzschild radius of a black hole, the “tilt” will be such that there is always a possible future trajectory that will allow a test particle to escape away from the black hole, though this will become increasingly difficult the closer we get to that radial point. At the Schwarzschild radius itself, the light cone is tilted such that its wall ( which is the surface of light ) coincides with the radius – there no longer is an escape route from the black hole, and even light itself will become “trapped” on the Schwarzschild radius. It can – at least in principle – remain here, but it cannot escape out to infinity, since such a trajectory is not inside the light cone of any event on the Schwarzschild radius; we say that an event right on this radius is not causally connected ( in the future ) to any other event outside the black hole. It is connected in the past though, as of course particle can get there from the outside – they just can’t get back out. Inside the Schwarzschild radius then, all light cones are fully oriented towards the singularity – not only is it not possible to escape back to infinity, but it is also impossible to remain stationary at a certain radial distance. The situation looks like this [21] :

The Schwarzschild radius hence forms the boundary that separates future-oriented events that are causally connected to other events outside the black hole, from those events that are not. It is a one-way portal – so long as you are outside this radial point, there are always ways to escape, but once the point is reached or passed, there simply is no way back out. This is why, for the case of black holes, the Schwarzschild radius is called its event horizon.

Event Horizon. An event horizon is a surface at which light cones become tilted to such a degree that no future-oriented trajectories lead back out to infinity. It is a one-way surface, in that events above and below the horizon can be causally connected only in the past light cone ( i.e. for an infall ) of the lower event.

Below the event horizon, due to how light cones are oriented, there cannot be any stationary or outward-moving observers, no matter how they fire their rocket thrusters; ageing into the future here means that you necessarily get closer to the central singularity. Formally, the radial and time coordinates in the metric trade their signs – the radial coordinate now becomes time-like, and the time coordinate becomes space-like. Ageing into the future means you move radially downwards -it is impossible to do anything else, because all light cones point “down”. You can delay and slow your in-fall, but you can’t stop it, unlike is the case for a region outside the event horizon, or for a region outside a regular body [22] :

Metric. Below the event horizon, the radial coordinate becomes time-like, and the time coordinate becomes space-like. Ageing into the future means you move towards the central singularity – no other trajectories are possible.

It is not just the orientation of light cones that changes as the event horizon is approached and intersected, but also their shape. An event horizon’s shape reflects the “spread” of all possible future-oriented trajectories emenating from an event – if its broad and flat, there are many possible routes, if it is long and narrow, there are very few possibilities [23] :

The closer you get to the horizon, the harder it becomes to move away from it radially ( the narrowing of the light cones ); inside the horizon, the opposite is the case, and the closer you get to the singularity, the harder it becomes to move in any direction other than radially towards it. The “flip” in orientation at the horizon in the above figure represents the change in character of the radial coordinate from space-like to time-like.

Light Cones. Light cones are tilted radially and compressed in the other directions as you approach the horizon – the closer you get, the fewer “escape routes” are available to a test particle. Below the horizon, all geodesics lead to and terminate at the singularity.

In terms of Schwarzschild coordinates, the event horizon is a coordinate singularity – the coordinate system is not defined at this point, and the region below it is not future-causally connected to any event outside of it. Let us take a closer look at what this mathematical fact physically means for an external observer.

Suppose we have a stationary observer very far away from the black hole; he takes two clocks, one of which he keeps in his hand, and the other one of which he decides to drop radially ( i.e. no angular momentum ) into the black hole. What will happen ?

To answer this, we need to remember that what is under consideration here is not just purely spatial distance on its own, or elapsed time on its own, but the full separation between events in spacetime. Just integrating radial distance is hence not enough – we also need to account for the non-trivial changes in temporal relationships between points on the infall trajectory. To do this, we hence must obtain an expression for radial velocity from the equation of motion (19) for a free fall without angular momentum; the result is

(39)

Because there is no angular momentum, and the clock is a massive test particle, the effective potential simplifies, and the radial velocity becomes

(40)

Note that this is proper radial velocity, meaning the change of the radial coordinate with respect to proper time, being the time that the falling clock physically records. Now that we know the radial velocity, we also need the physical ( =proper ) distance between observer and event horizon; this is obtained by evaluating the integral (24) :

(41)

This physical proper radial distance from observer to event horizon is finite for all . We are now in a position to calculate the total proper time which the clock will record as it falls the radial distance (41) with radial velocity (39) :

(42)

where we have set all constants to unity for simplicity. This result is again finite and well defined for all . This leads us to our first conclusion about free fall into a Schwarzschild black hole :

Free Fall ( Falling Observer ). An observer freely falling from rest into a Schwarzschild black hole will reach the event horizon after travelling a finite proper distance as measured by his own rulers, and in finite proper time as measured by his own clock. His infall velocity with respect to a stationary reference point just above the horizon approaches the speed of light.

So far so good. But what about a stationary observer far away from the black hole ? What will he see ? To answer this we simply take our results from above, and recast them in terms of coordinate time, as opposed to proper time. The physical distance (41) to the horizon still remains the same – it is finite for an external observer just as it is finite for an observer in free fall. What does change though is the radial velocity, and hence the total infall time; because proper time and coordinate time are related as ( see the metric )

(43)

the integral (42) picks up an extra term and becomes

(44)

Hence, the time that a distant observer measures the infall to take is infinite, because radial coordinate velocity ( see (13) ) becomes zero at the horizon. In other words, for a stationary observer far away, nothing ever reaches the event horizon.

Free Fall ( Stationary Far-Away Observer ). An observer who drops a test particle into a black hole will never see or calculate it to reach the horizon. The infall time becomes infinite for him. This is because – while the distance to the horizon remains finite -, the radial velocity he calculates goes to zero as the test particle approaches the horizon; it “freezes” in place just above it.

So, a freely falling observer reaches the horizon in finite time after falling a finite distance, yet a stationary observer far away never sees anything reach the horizon at all. Isn’t that a contradiction, a paradox ? No, because we are not comparing like-for-like here; the infalling observer and the far-away observer do not share the same notions of “time” and “simultaneity”, so their disagreement on what happens here is both expected and necessary. The far-away observer remotely analyses the situation based on clocks and rulers that are far away, whereas the infalling observer physically measures everything with instruments that are local to the infall world line itself; in essence, he directly measures the physical length of the world line, and directly traces its physical trajectory in space. The remote observer does not; his point of view is one of someone who projects what physically happens somewhere far away in curved spacetime into a patch of locally flat spacetime; his view is hence necessarily distorted, because his system of coordinates does not faithfully represent distances, or separations in time. It’s like projecting the surface of a sphere onto a flat plane – it can be done, but the result will be grossly distorted as compared to the actual, physical appearance of what is on the sphere.

Because an observer in free fall faithfully “follows” the geometry of its actual world line in both space and time, and because the length of that world line is a generally covariant quantity ( which is finite and well defined ), all observers agree on what he measures. On the other hand, the distorted view of the far-away observer is also valid, but only in his local, far away frame of reference.

Free Fall World Line. An observer in free fall traces out a geodesic in spacetime from somewhere far away to the horizon and beyond. He follows the spatial trajectory of the world line, and his clock directly measures the physical ( geometric ) length of the world line in spacetime. All observers agree on the world line’s geometry, including far away observers – this is therefore an acurate description of what physically happens to a test particle in free fall. However, what a far-away observer sees on his own clock ( which is remote to the actual world line itself ) is different, and is valid only is his own remote frame.

Since free fall observers and far-away stationary observers do not share the same concept of time, there is necessarily one other quantity which they must disagree on, and that is energy. Specifically, for the case of free fall, energy at infinity ( the far-away observer ) is related to energy on a local shell close to the black hole via the relation

(45)

One immediate consequence of this is that, if you have a light source deep inside a gravitational field emitting photons towards a far-away receiver, then that receiver will see this light redshifted. This phenomenon is called gravitational frequency shift, and a manifestation of the fact that, in curved spacetimes, energy is an observer-dependent quantity since the light cones of different observers are tilted with respect to one another [24] :

In physical terms, this means that, if a far away observer drops a test particle in to a black hole, then not only will he see it slow down as it approaches the horizon, but he will also see it increasingly redshifted and growing dimmer.

Gravitational Frequency Shift. In curved spacetimes, energy is an observer-dependent and local quantity; a photon travelling through an extended region of curved spacetime will be frequency shifted.

The last effect we need to talk about before we can bring it all together into a coherent picture of what an infall into a Schwarzschild black hole is like, is tidal gravity. If you place an extended ( i.e. not point-like ) body under the influence of gravity, then, due to his spatial extension, different parts of the body will experience different amounts of gravitational “pull”, leading to induced tidal forces. Relative acceleration between test particles follows from the Riemann curvature tensor; for the case of radial infall in Schwarzschild spacetime, the relevant components are

(46)

and

(47)

Physically the above means that bodies are stretched along the radial direction, and compressed perpendicularly to it [25] :

The Riemann curvature tensor is a covariant quantity, so all observers agree on the tidal character of spatial curvature. Hence, a far-away observer will see a dropped test particle slowing down, growing dimmer and more redshifted, and being stretched and compressed by tidal forces. Putting it all together, here is an animation of how it would look like – turn up your speakers, click on “Begin Experiment” and “Drop The Clock”, watch, and listen [26] :

Things are somewhat different for someone who physically falls into the black hole himself; here is what it would look like from his point of view [27] :

As we have found, the infalling astronaut will reach the horizon in finite time as measured by his own clock, so his world line is of finite length, and the horizon is at a finite distance from him. The strength of the tidal forces he experiences depends only on the total mass of the black hole, and drops off according to an inverse cube law – if the black hole is large and very massive, tidal forces are almost inconsequential until just before the singularity is encountered. For small, very light black holes, tidal forces would rip our astronaut to pieces long before he even gets close to the event horizon.

What the above ( very simple ) animations do not accurately represent is the view of the outside universe as you fall in; hence I will wrap things up by giving you a full video that manages to do this as well [28] :

As you can see in this video, there is still a lot more to be said about black holes; there are a lot of things I have not yet spoken about, or hinted at only briefly – such as anti-horizons, wormholes, gravitational lensing etc etc. However, since this article is only meant as a first introduction to the subject, it makes little sense to try and cram everything into it. I will soon revisit the subject of Schwarzschild black holes in a separate article, which will provide a more in-depth view.

To briefly sum up, here is what we have learned :

1. Schwarzschild spacetime can be understood as a family of nested round spheres
2. Schwarzschild coordinate accurate represent spherical symmetry on each sphere, but does not accurately reflect distances and separations in time
3. Schwarzschild coordinates are an “accounting device” ( bookkeeper coordinates ) and do not represent any real, physical observer. All real-world measurements must be computed from the metric
4. At the Schwarzschild radius, the coordinate system becomes singular, but physical spacetime remains smooth and continuous
5. In Schwarzschild spacetime, energy per unit mass and angular momentum per unit mass, taken at infinity, are constants of motion
6. Orbits and trajectories of test particles in Schwarzschild spacetime differ from their counterparts in Newtonian gravity in several aspects; this gives rise to phenomena such as light deflection, perihelion precession, and the Shapiro effect
7. It is important to distinguish between coordinate and proper quantities – the former are merely a bookkeeping system, whereas the latter correspond to physical measurements
8. The interior metric describes an object in equilibrium between its own gravity, and its internal pressure. Once certain limits are exceeded, there is no known mechanism to provide sufficient counterpressure, and the object will undergo gravitational collapse
9. A region of geodesic incompleteness shielded by an event horizon is called a black hole
10. Singularities are not actual, physical objects, but rather a sign that we are operating outside GR’s domain of applicability, so the theory cannot make sensible predictions there
11. Gravity tilts light cones
12. Time and space “trade places” at the event horizon
13. Stationary far-away observers never see anything reach the horizon, whereas freely falling observers get there in finite time after traversing a finite distance

Stay tuned for my next blog post, where we will be diving a little deeper into the physics of black holes !

#### References :

[1] Edwin Taylor & John Wheeler, Exploring Black Holes, Princeton University Press

[2] http://www.anti-dialectics.co.uk/Spheres.jpg

[3] Edwin Taylor & John Wheeler, Exploring Black Holes, page 2-26, figure 7

[4] http://pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_massive/circles_2.png

[5] http://eagle.phys.utk.edu/guidry/astro616/lectures/lecture_ch18.pdf, figure 18.2

[6] Edwin Taylor & John Wheeler, Exploring Black Holes, page 2-28

[7] http://scienceblogs.com/startswithabang/files/2014/08/image001.gif

[8] Edwin Taylor & John Wheeler, Exploring Black Holes, page 4-23, figure 11

[9] Edwin Taylor & John Wheeler, Exploring Black Holes, page 4-9

[10]Edwin Taylor & John Wheeler, Exploring Black Holes, page 5-6, figure 2

[11] http://pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_massive/deflection.png

[12] http://pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_massive/deflection_Schwarzschild.png

[13] http://pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_massive/Mercury_1_anim.gif

[14] http://pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_massive/Mercury_2_anim.gif

[16] http://eagle.phys.utk.edu/guidry/astro616/lectures/lecture_ch18.pdf

[17] Misner/Thorne/Wheeler, Gravitation, page 641, fig 23.1

[18] Misner/Thorne/Wheeler, Gravitation, §23, box 23.1

[20] http://pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_massive/tipped_cones.gif

[21] http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/black_holes/bh_lightcones_st.gif

[22] http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/black_holes/grav_coll_st_diag.gif

[23] http://i.stack.imgur.com/AiiZF.jpg

[24] http://pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_massive/red_shift.gif

[25] http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/black_holes/tidal_forces.gif

[26] http://hubblesite.org/explore_astronomy/black_holes/encyc_mod3_q15.html

[27] http://hubblesite.org/explore_astronomy/black_holes/encyc_mod3_q16.html