Black holes and Schwartzschild geometry

First some questions and some obvious answers for the reader:

· How long is a second? One second is the time that elapses during 9.192631770 x 10⁹ cycles of the radiation produced by the transition between two levels of Cesium 133.

· How long is a meter?One meter is the distance traveled by a ray of electromagnetic (EM) energy through a vacuum in 1/299,792,458 (3.33564095 x 10^-9) second. don't believe me? check this out

· What is the shortest path between two points? The straight line

· What is the curvature of a straight line? Zero

· Do parallel lines intersect each other? No

To find other answers to these questions than you would expect, take the courage to click on each link.

Black holes are very exotic objects, in the way that we cannot apply common sense to deduct the behavior of anything which is close to a black hole. This however makes them all the more interesting.

If common sense does not help us, we must use mathematical formulas and simulations to “see” what happens. The mathematical formulas are provided by Einstein’s famous general theory of relativity. In essence, it reformulates the laws of gravity in terms of space-time geometry. The principle of equivalence the starting point of general relativity, states that in any local (that is, sufficiently small) region in space-time, it is possible to formulate the equations governing physical laws such that the effect of gravitation can be neglected. Take, for a concrete example, the astronauts in a satellite orbiting the Earth. They are constantly under the influence of gravity. But yet, in the satellite station, things behave as if there was no gravity, you see on TV how cups float in the air and so on.

Eliminating gravity from the picture, something else had to be changed to account for its influence.

newton vs. einstein

SOME FACTS ABOUT CURVED GEOMETRY

To help the understanding of the new approach to gravity and the universe, scientists often resort to embedding diagrams, which show curved surfaces as examples of Einstein’s curved geometry. Of course not all 4 dimensions (3 spatial and 1 temporal dimension) can be drawn on paper (unless you do it artistically, as the picture on the left…). Often, these diagrams suppress one or two spatial coordinates, to give a clearer picture. There can be three types of curved surfaces, according to the value of the curvature: positive, negative or zero. The embedding diagrams for these types of surfaces look similar to these:

We have been referring to “curved geometry”, “curvature” and so on, but what is curvature? In Euclidean geometry, the term refers to the inverse of the radius of the circle tangent to the surface. Here, curvature is given a physical sense:

Zero curvature

Positive Curvature

Negative Curvature

It refers to the trajectories in space-time of two rays of light, which start out parallel to each other. Do parallel lines intersect? Not in flat space, on which the Euclidean geometry we learn in school is based. But in positively curved spacetime, they do, while in negatively curved spacetime, they get further and further apart. This should give you a sense of what curvature refers to.

Another way of looking at curvature (this time considering also its value, not only the sign) is to think of GEODESICS. A geodesic is the shortest path between two events, or points in 4D. What is the shortest path between two points? On the curved surface, a straight line which has every point contained in the surface simply does not exist. (Try drawing a straight line on a sphere!) Depending on how curved a surface is, the geodesics get more and more curved, thus longer and longer compared to the straight line connecting the points.

DOWN THE BENT AXIS, AS DEEP AS WE CAN GO

Now that we went through all the trouble to explain this new geometry, how does it relate to black holes? As all massive bodies, black holes produce curvatures in space-time according to Einstein. How? First of all let us see how a massive star curves spacetime. The most widely used analogy is that of a heavy ball placed on a rubber sheet (though it can be misleading in many ways):

rubber sheet analogy

space bending

The curved spacetime around a massive body looks similar to this, although one spatial coordinate is replaced with the temporal one.

To properly and quantitatively describe events in curved spacetime, one would like to have a metric, a recipe to show one how to work with the four coordinates. If you are familiar with the scalar product of two vectors, a metric is easily explained: it gives the scalar product of two four-vectors, that is vectors with 4 components instead of 3. One of the biggest mathematical challenges was to modify the special relativistic metric to suit gravitational influence, to take into account the new geometrical constraints posed by general relativity. The first solution was offered by the German scientist Karl Schwarzschild, who assumed a spherical symmetry of the solution in order to simplify the equations. Due to spherical symmetry, he transformed the special relativity invariant from Cartesian to polar coordinates:

ds² = dr²+ r²df²- c²dt²

Then, he assumed a more general invariant would suit the General Theory of Relativity… Makes sense, right? So he added two coefficients to dr and dt like this:

ds² = A*dr²+ r²df²– B*c²dt²

Now, the challenge was to determine A and B using Einstein’s field equations. The result, known as the Schwarzschild metric, is

dt²= (1-2M/r)dt²– dr²/(1-2M/r) - r²df²

where

Ø M is shorthand for GM_star/c²

Ø r is the coordinate from the center of the black hole. An observer situated near a black hole can compute this coordinate by “walking” along a circle with the center at the singularity, measuring the circumference of this circle and dividing it by 2p

Some interesting facts immediately visible from this metric:

Ø look at the coefficients of dt and dr. They depend on r. Think of dr as an infinitesimal length; for all intended purposes 1cm is small enough compared to r, so take dr=1m. Now suppose you have an observer near the black hole, freely falling towards it at some instant t, and another observer at rest, far away. The first observer is holding a stick 1m long, parallel to his direction of motion. How long will this stick appear to the far-away observer?

where dr_shellis one meter and dr id the length measure by the far-away observer. How long is a meter?

Ø Even more interesting facts result from the following: suppose the observer falling into the black hole measures his displacement between r₁ and r₂. You would expect him to find that this displacement is equal to r₁-r₂. That’s because you are thinking in terms of flat spacetime! Indeed, an observer far away will (where spacetime is flat) will see that our freefalling observer has moved exactly r₁-r_2.The freefalling observer however experiences the curvature of spacetime; he is not moving along a straight line, but along a geodesic. The distance he measures is surprisingly NOT equal to the difference of his initial and final coordinates. It is

Ø Similar to length variations, time variations appear:

How long is a second? The most important consequence of this is the gravitational redshift: a photon with a certain frequency relative to an observer near the black hole (who measures dt_shell) will have a shorter frequency when seen by a far away observer.

Ø Another question immediately emerging was: what happens at 2M=r? What if the star shrinks to a radius smaller than this point? Then the coefficient in front of dt would be zero at r=2M, whereas the coefficient of dr would be ¥. There is something special about this r=2M. It is the radius of the event horizon of a black hole! (more about event horizons from Simonetta) We already saw how the declining radius of stars influences space-time curvature. What happens as r->2M?

The curvature at the event horizon goes to infinity!. This is the very characteristic of a black hole. What does it imply? First of all, what you already know about black holes: no light can escape. To see this, we superimpose on the diagram above “light-cones”. A light-cone is also a 3D surface with time as the vertical axis and 2 of the spatial coordinates on the other axes. The geodesics of this surface show the possible paths of light in space-time.

Here is an example:

A light cone

lightcones

Let us first consider a 2D representation: by simply taking x = ct, we see that in normal (not curved) geometries, a graph with t on the vertical axis and x on the horizontal axis will simply be a line passing through the origin for t=0 (the “present”). Rotating this line around the time axis to include a second spatial coordinate in the picture, we get the cone above.

What happens now to light-cones around a black hole? (Red half of the cone is its “past”, green is its “future”).

Far away, space-time is essentially flat, so no effect is felt. As we move closer, light-cones begin to tilt, as space and time intervals are modified according to Schwarzschild’s metric. Finally, closer and closer to the event horizon, light cones are so tilted that no geodesic on the cones points towards the “outside;” all the light which is emitted unavoidably heads towards the center of the black hole, the singularity. Notice also that the axis of the cones, which started out as vertical – time axis – becomes almost horizontal nearer to the horizon. But horizontal axes represent spatial coordinates on our diagram! The roles of time and space are interchanged! While in our world you cannot stop time, beyond the horizon of a black hole you cannot “stop” space, it is impossible to maintain the same coordinate.

WE HOPE YOU ENJOYED OUR JOURNEY

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Image from: properties of time

Acknowledgements and links you might want to take a look at if you have read my page up to this point