Particle Paths Near Black Holes

Hello!  My name is Ashley Stock and this summer I had the privilege of being a summer student at MSSL, under the supervision of Prof. Kinwah Wu.  I investigated the motions of massless particles (photons) in close proximity to non-rotating (Schwarzschild) and rotating (Kerr) black holes.

With recent announcements of gravitational wave detection of coalescing black holes by LIGO, and other observational evidence, it is becoming increasingly clear that black holes can exist in real, astrophysical settings and are not just mathematical curiosities.  It is therefore important for us to understand how these objects influence their surroundings and the observational consequences of these influences.

In general relativity, the geometry of space and time is defined by the matter which is in the space time.  This geometry then defines the motion of the matter.  In the absence of any non-gravitational forces, a particle will travel along the shortest path (a ‘geodesic’).  In flat space time this is a straight line, but when space time is curved as it is near black holes and other massive objects, the paths will become curved or bent.  Additionally, the energy of the particles will change as they travel in the space time.  In trying to escape a black hole, a particle will lose energy to the gravitational field.  This is referred to as ‘gravitational redshift’ because light which loses energy becomes redder.  The particular space-time geometry for a static black hole depends on its mass, spin, and charge.  In this project, only uncharged black holes were considered.

There are many other ways that light can be affected as it travels from its source to an observer.  If the source rotates parts of it will be moving towards the observer while other parts are moving away.  The motion of the source effectively pushes the waves of light closer together to make it bluer, or pulls the waves of light farther apart to make them redder.  Therefore, light emitted from the parts moving towards the observer will be bluer and parts moving away from the observer will be redder.  This effect, known as the ‘Doppler shift’, along with the gravitational redshift, changes the colours of the light we observe in comparison to the colours emitted.  There can also be objects between the source and the observer which can absorb, scatter, or emit light.  This will affect the intensity (or brightness) of the light observed.

Based on code previously developed by Ziri Younsi, I was able to calculate the intensities and redshifts of light emitted from a thin disk of material orbiting around a black hole (an accretion disk) to create images of the accretion disk as seen by a distant observer.

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Figure 1:  Observed frequency shift of light with logarithmic scale emitted from thin accretion disk of non-rotating (top) and maximally rotating (bottom) black holes.  Inclination angle is increased from 1 degree to 89 degrees.

 Spin0InclinationEvolution189 Spin099InclinationEvolution189

Figure 2:  Observed intensity of light with logarithmic scale emitted from thin accretion disk of non-rotating (top) and maximally rotating (bottom) black holes.  Inclination angle is increased from 1 degree to 89 degrees.

Some interesting things to notice in these images are the shapes, sizes, and colours.  For low inclination angles (which correspond to looking straight down onto the face of the disk), the shape is similar to what you might expect.  As the inclination angle increases, the shape appears increasingly warped.  The gravity very near a black hole is sufficient to completely change the direction of light, so it is possible that light emitted from behind the black hole or the bottom of the disk can still reach an observer.  This can mean that the light appears to come from somewhere completely different from where it was actually emitted.  Light emitted from one part of the disk can also hit another part of the disk before reaching the observer contributing to ‘higher order images’–where the order of an image is the number of times the light has hit the disk before reaching an observer.  To create the images in Figures 1 and 2, we summed the light from images up to order 5.  Higher order images become increasingly dimmer, but can contribute to different observed features (as shown in Figure 3).  For the lower inclination angles, it is harder to see the first order images because they still appear to come from the top face of the disk (although they are actually coming from the underside of the disk) but they contribute to observed intensity.

The accretion disk used has an innermost radius at the innermost stable circular orbit (ISCO).  This corresponds to a radius where free falling matter can orbit around the black hole in a circle without falling into the black hole.  The radius of the ISCO depends on the spin of the black hole, and is smaller for greater spin.  A similar region exists for light.  Known as the photon sphere, this is the region where light has a circular orbit around the black hole.  The radius of the photon sphere is also dependent on the spin of the black hole, with a smaller radius for greater spin.    The inner radius of the circles that can be seen around the centre of the images from non-rotating black holes (top images) corresponds to the radius of the photon sphere.  With increasing spin, the ISCO becomes closer to the photon sphere and the light from these regions cannot be distinguished in the maximally rotating case.  The amount of gravitational redshift observed depends on the closest the light gets to the black hole’s centre.  Since the photon sphere is smaller for greater spin, light can get closer to rotating black holes, and therefore experience greater redshift.  This can be seen in Figure 1, where the light near the centre of the rotating case is shown redder than other parts of the disk.

As the inclination angle increases, Doppler shifting becomes more important because it depends on the speed of the disk directly towards or away from the observer.  You can therefore tell what direction the disk is spinning because the side which is redder is the side moving away–for these images, that means the disk is spinning counter-clockwise.  The rotation of the disk and black hole also have another effect, relativistic beaming.  A source which is going very quickly, emits more strongly in the direction it is moving.  So although it may appear that there is more stuff to the left of the rotating black hole, it is an optical illusion caused by light that might otherwise be directed away from the observer being directed towards on the left, and light that might otherwise be directed towards the observer being directed away on the right.


Figure 3:  Observed intensity of light with logarithmic scale emitted from thin accretion disk of non-rotating black hole for different image orders, with a viewer inclination angle of 45 degrees.


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