Viewing Gravity

Introduction

An example of a rotating black hole

Introduction

The fundamental insights from relativity can be stated without using the advanced mathematics (differential geometry or tensor calculus) that underlie the formal articulation of the theory. Nevertheless, these concepts describing the fundamental properties of relativity are difficult to ponder given their abstract nature. For example, space and time are best described as a single four-dimensional continuum, spacetime; energy, momentum and stress all warp this spacetime fabric; and bodies move through this curved spacetime by following the closest analog to a straight-line path.¹

This page is part of an ongoing effort,² which has been given the name “Viewing Gravity” to use visualizations to investigate and convey the nature of gravity in general relativity as a way to circumvent the difficulties of mathematics and the limitations of words in understanding the nature of general relativity. It builds upon several papers described in a research effort which is summarized in the following post on this website (here). It also relies upon modifications to the visualization method described in that post. These are summarized on a page on this website (Visualizations of Spacetime Curvature), and this new method made animating the spacetime geometry much more tractable.

The visualizations aim to highlight the two key physical effects of curved spacetime that cannot be eliminated through a judicious choice of measuring lengths or times (in a more technical language, setting up a locally inertial reference frame). First are relativistic tidal effects, which stretch an object in one direction and squeeze in two perpendicular directions (or vice versa) via a relative acceleration. The new visualization method highlighted how these stretching and squeezing effects change throughout space by showing them as red (stretching) or blue (squeezing) line segments distributed throughout space; the relative size of the icons at each point indicated the scale of the stretching and squeezing at each point, and a translucent color showed the relative magnitude of the stretching and squeezing effects at different points. Second is a relative rotation effect, which quantifies how space gets dragged into motion compared with nearby points (referred to as frame dragging). Its effects are visualized using the same method as with tidal effects, but the directions now refer to the axis of rotation at each point (with red denoting counterclockwise rotation and blue clockwise), and the translucent colors indicate the size of the relative rate of rotation at different points.

An example of a rotating black hole

To illustrate a specific example of this visualization method, the method was applied to a black hole rotating at ninety percent of the maximum allowed value. The animations shown below were created by former UVA undergraduate physics major Zachary Raney (‘25). The black hole is rotating at a uniform rate and is symmetric about its axis of rotation; consequently, the spacetime does not change with time. Its event horizon (the surface from which not even light can escape) is a black, squashed ellipsoid in the coordinates used in the animations below. However, to give a better view of how the spacetime appears at different distances from the event horizon of the black hole, the animations use a moving viewpoint to highlight the regions of space close to and further away from the event horizon. Two separate visualizations below show the tidal effects and frame-drag effects, respectively. For both types of effects, their magnitudes become weaker as the distance from the event horizon increases. However, the amplitude of the fields were scaled at each radius in the animations to remove this weakening so as to highlight the angular dependence of the tidal and frame-dragging effects.

The first of the two videos shows the tidal field. The regions of blue-green color show that the magnitude of the stretching and squeezing effects of the spacetime curvature close to the event horizon are somewhat stronger near the equator rather than they are along the axis of the black hole’s rotation (which points upward in the animation). Further from the event horizon, the angular dependence becomes less pronounced and more uniform in the angular directions. In addition, the red icons showing the positive (stretching) relative acceleration point predominantly radially away from the center of the black hole’s horizon. This is expected because the spacetime becomes more strongly curved closer to its event horizon. The squeezing direction are oriented primarily in the tangential directions. However, because of the dragging of inertial frames the stretching direction has a small twist about the black hole’s axis of rotation.

The second animation is of the frame-dragging field of the rotating black hole. The blue conical regions centered on the axis of rotation around the poles are where the strongest relative rotation effect occur. Although the regions are given the same color, the sign of the relative rotation differs between the northern and southern poles. This can be seen more clearly from the icons as the relative rotation has one clockwise direction in the northern hemisphere and one counterclockwise direction in the southern hemisphere. The region in the equatorial plane has the weakest relative rotation effect, which is indicated by its light green color. This is required by the fact that the sense of relative rotation has the opposite sign in the two hemispheres.

1. John Archibald Wheeler famously summarized the last two aspects of the theory as “Spacetime tells matter how to move; matter tells spacetime how to curve” assuming, of course, the reader was familiar with the notion of spacetime. ↩

2. This material on this webpage was based upon work supported by the National Science Foundation under the NSF CAREER Award PHY-2439893. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. ↩