10: General Relativity
- Page ID
- 30487
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Chapter 10 presents Einstein’s General Theory of Relativity, in which the force of gravity is a natural consequence of curved spacetime due to the presence of mass and energy. You will investigate properties of curved spacetime including time dilation, gravitational redshifts, and gravitational lensing. You will also build a mathematical construct for describing curvature and apply it to two- and three-dimensional models.
- 10.0: General Relativity Introduction
- If you did not know beforehand, is there any way that you would be able to determine that you were on a rocket ship and not, say, sitting in a windowless room in a building on Earth? Is there any sort of experiment that you could do that would distinguish these two cases for you? The answer to that question might surprise you, and it will lead us to a remarkable new view of gravity, just as it did Albert Einstein more than 100 years ago.
- 10.1: Einstein’s Equivalence Principle
- Recall that special relativity deals with unaccelerated motion. Its natural extension, therefore, is to try to describe what happens when motion (or frames of reference) are accelerated. Such a theory would be more general than one limited to uniform motion. It might therefore be called the general theory of relativity — and so it is.
- 10.2: Gravity and Curvature
- In this section we will explore some of the consequences of the weak equivalence principle. Let us again consider a rocket ship far away from any source of gravity. Earlier, we imagined the motions of material objects. Now we will consider what happens when light passes through the rocket ship under these same conditions.
- 10.3: What is Curvature?
- The distortion of spacetime is more complicated in general relativity. We say that spacetime is flat and static in special relativity. In contrast, spacetime is curved and dynamic in general relativity. What is meant by flat and curved? In this section you will explore the meaning of these words in some detail in order to help you better understand what Einstein’s view of gravity is really all about.
- 10.4: Tests of General Relativity
- In this section we will discuss some of the observational tests of the General Theory of Relativity. The theory makes a number of predictions about the world that are distinct from those of Newtonian gravity. We have already looked at several. Here we will consider a few more, and in somewhat greater detail. A number of these topics will be covered in later chapters, and this section serves as an introduction for deeper investigations to come.
- 10.5: The Source of Gravity
- In the descriptions of gravity we have looked at so far, we have left off one important part: What is the source of spacetime curvature? We have described what curvature is, at least in a basic sense, and we have described how curvature affects the motion of particles and the perceptions of space and time that different observers have. But where does the curvature actually come from? You will understand that the curvature of spacetime is related to the mass-energy distribution.
- 10.6: Wrapping It Up 10 - Curved Spacetime Around Astronomical Objects
- Imagine you are studying a binary system in which a pulsing neutron star is orbiting a supermassive black hole and you are viewing the system from the direction of the black hole. Over time you have detected that the time interval between the pulses from the pulsar is changing. Intrigued, you set about to discover what is happening by studying a graph of the pulsar’s periods.
Thumbnail: Artist concept of Gravity Probe B orbiting the Earth to measure space-time, a four-dimensional description of the universe including height, width, length, and time. (Public Domain; NASA).