The Geometry of a Black Hole

were often told in physics that the presence of massive bodies warps the fabric of space and time but what does such a statement actually mean most visuals are heuristic tools that don't offer clear understanding leaving one to wonder is it possible to actually visualize space-time curvature especially near an object so massive we can cross a limit of no return this is dialect with the geometry of a black hole what does the curvature of space and time around a massive body actually look like to answer this question we need to turn to a mathematical object called the Schwartz Shield metric presented in tensor form this metric is a matrix with four Expressions along its diagonal components now in our prior videos we demonstrated how the components of a metric tensor can be interpreted as instructions for resizing infinitesimal pieces of a map in order to fit those pieces to a curved surface and indeed this is the case whether your map is one of a physical surface like that of the Earth or whether it's a map of a more abstract surface like that of the space-time manifold we can thus use the Schwartz Shield metric to construct for ourselves a model of curved space-time but first we must get our hands on the map which the metric belongs to here we turn to what's formerly known as the spark Shield coordinate chart this map depicts space-time in the vicinity of a single centrally located non-rotating uncharged spherical Mass now just as there's no one right way to map out the curved surface of the Earth onto a two-dimensional Cartesian grid there's no one right way to map out the curved surface of space-time onto a four-dimensional Cartesian grid consequently we have a choice in how to make such a projection motivated by the spherical symmetry of the region the Schwartz Shield approach is to assign to every world event that occurs a set of more or less spherical like coordinates t r Theta and Phi the first coordinate T is intended to be analogous to a traditional time coordinate as there is no Global time in this region and clocks everywhere tick at different rates the simplest solution for describing time here is simply to choose a Master Clock meaning a clock against which all others can be compared the Natural Choice will be that of a clock belonging to a distant Observer since far out from the Central Mass space-time curvature will be negligible all events which transpire across this space-time region are thus assigned a Time coordinate based on whenever this distant Observer perceives them to occur foreign the second coordinate R is analogous to a radial coordinate it is produced by placing observers in concentric Rings about the mass then having those observers measure the circumferences of the Rings in order to deduce via dividing by 2 pi what the respective radii of the Rings would be were the overall geometry of the space-time region there flat this is like making a bird's eye map of a circularly symmetric Hill the radial distance between any point and the top of the hill obviously depends on the Hill's curvature and grade but once projected onto a flat surface this distance instead becomes the circumference of the Contour which the point occupies divided by 2 pi meanwhile the third and fourth coordinates Theta and Phi retain their ordinary meaning from spherical coordinates as the polar and azimuthal angles about the Central Mass together these four coordinates once assigned to every event map out with time and space would look like in this region if space-time were flat but of course space-time isn't flat here it's curved which is why once we feed our map into Einstein's equations out pops the schwarzchil metric telling us how our artificially constructed coordinates relate to the true space-time geometry now since the last two components of this metric are the same as they would be in flat space time for the time being it'll be simpler to drop them from the picture and only consider a two-dimensional space-time manifold don't worry though we'll add those Dimensions back in later we are now ready to transform our map I.E the smart Shield coordinate chart into an actual space-time surface the procedure is simple to start we cut the map into many tiny Square pieces Dr by DT then we simply follow the instructions from the metric tensor now the first instruction tells us by how much we need to stretch or Shrink the coordinative DT length of each map piece in order to match it to its true surface length I.E that length of proper time an observer located there would experience here that value corresponds to the square root of 1 minus 2gm divided by c squared times the r coordinate of that piece foreign we can see that for very large R coordinates this expression takes on a value of nearly one meaning that the temporal lengths of pieces far from the Central Mass will remain essentially unchanged for smaller R coordinates however this expression takes on values increasingly less and less than one meaning that the temporal lengths of pieces closer and closer to the Central Mass will be more and more shrunk physically this means that from the perspective of the distant Observer clocks closer to the Central Mass will appear to run more slowly than clocks farther away next the second instruction from the metric tensor tells us by how much we need to stretch or Shrink the Dr length of our piece in order to match it to its true surface length I.E the length of proper distance an observer located there would measure here that value is just the reciprocal of the former expression meaning anywhere we saw temporal lengths decrease spatial lengths will increase by the same factor pieces far from the Central Mass then will just as with the temporal case see their spatial lengths remain relatively unchanged while closer to the mass pieces will see their spatial lengths increasingly stretched foreign physically this means that from the perspective of a distant Observer rulers closer to the Central Mass will appear more and more contracted we now have all the correctly resized pieces of our space-time map meaning we are looking at our spatio-temporal reality in its true proportions at this point let's draw in proper space and time grids on each piece this will help us understand how the flow of time and space will look across different points of the manifold now although we lack a meaningful higher dimensional coordinate transformation with which to appropriately reassemble these pieces thanks to the fact that we have suppressed the other two dimensions of space we still have a third dimension left over to play with although we can't interpret this extra Dimension as anything real there's no reason we can't use it to Aid in our visualization either so let's put the puzzle together the trick here is to First roll the entire surface up in a cylindrical fashion and then to bend the tapered edges in towards one another until adjacent pieces are all touching and voila there we have it what the curvature of space-time in the vicinity of a massive object actually looks like now note here that where the surface curves around it doesn't actually rejoin itself rather as we move along the temporal direction we can imagine the surface getting continually rolled up like a large sheet of paper also note that the boundary of the manifold lies at the edge of the Central Mass as the schwarzshield solution can't be extended to the masses interior lastly to make this picture more accurate we can repeat this process with smaller and smaller pieces until we essentially have approximated a smooth manifold next let us consider the motion of two distinct observers sitting together out at some coordinate radius r on this manifold one Observer is inertial or Free Falling and the other is accelerating in order to maintain constant radial distance from the mass over a very short period of time they follow the same world line together on the manifold but in the next instant the velocity of the accelerated Observer increases with respect to the inertial Observer and their world lines diverge now the proper grid on this next manifold piece depicts space and time as measured at a constant radial distance meaning it belongs to the accelerated Observer not the inertial Observer thus we must transform into the frame of the accelerated Observer which results in The inertial Observers World line being deflected inwards towards the Central Mass high up along the manifold where the curvature is weak the amount of this deflection will be very slight but further down it becomes more and more substantial foreign inertial or free-falling object starts out moving purely in the temporal Direction the curvature of the manifold soon helps redirect it down towards the Central Mass to see this better we can keep the object on a constant radial line and depict the progression of time as a rolling of the manifold as though it were being rotated about an axis running through its Center from this perspective as the manifold rolls forwards in time the object begins to fall faster and faster down towards the Central Mass next we can add our missing spatial Dimensions back into the picture to do this we first draw radial lines outwards from the mass then we attach a rolling manifold to each line the visualization gets a bit awkward here as we are superimposing multiple manifolds atop one another but next we can sort of squash out the temporal rolling leaving only a visual of the radial motions now we see that we have a three-dimensional space where all objects fall steadily inwards towards the Central Mass as time progresses I.E we have reproduced a classical gravitational field so we can see how the geometry of space-time helps create the illusion of gravity but what happens if the curvature of our manifold becomes so great that there exists a limit of no return well returning to our Schwartz Shield metric we see that for a certain value of the r coordinate R equals 2gm over c squared the temporal component of the metric goes to zero while the spatial one blows up to Infinity in terms of resizing our map this means that pieces near this R coordinate value will become extremely thin and long when constructing the surface then these pieces will ultimately converge at a single point sealing off the space-time manifold there this point is quite literally the point of no return the object's Event Horizon if this object is heavy and dense enough The Event Horizon will lie exterior to its mass and it will be dubbed a black hole let's again look at the motion of two observers but now near this manifold's event horizon as they cross from one instant to the next the world line of the free-falling Observer will be deflected by an amount almost equal to the world line of light foreign s The Accelerated observer's velocity has to increase by nearly the speed of light in order for them to remain at a stationary radial distance once the inertial Observer Falls past the Event Horizon their speed is essentially now faster than the speed of light at least relative to The Observers outside the Event Horizon meaning they are causally cut off from the rest of the universe so there we have it the geometry of a black hole guess it turns out our higher dimensional reality rather looks something like a bullet a joke from God perhaps well at the end of the day this is still only a visualization and consequently there are a few things that do need to be qualified about it as well as a few things that are incomplete about it but those matters require a deeper dive into both relativity and topology so stay tuned as we'll continue exploring the meaning of space-time curvature in future videos this has been dialect thanks for watching

Channel: Dialect

Views: 766,795

Keywords: general relativity, gravity, spacetime, curvature, physics, learning, education, science, manifold, tensor, metric, Schwarzschild, black hole, black holes

Id: q6RufF4a6LM

Length: 18min 31sec (1111 seconds)

Published: Sat Dec 17 2022