Today in Astronomy 102: general relativity and the prediction of the existence of black holes

1Astronomy 102, Fall 20012 October 2001

Today in Astronomy 102: general relativity and the prediction of the

existence of black holes Experimental

tests: Einstein’s theories are proven to be valid.

Schwarzschild solves Einstein’s equations, applied to stars, and finds black holes in the results.

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Image: gravitational lens in galaxy cluster AC114, by the Richard Ellis and Ian Smail with the Hubble Space Telescope. Note that the “quasars” A and B are mirror images of each other.

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2 October 2001 Astronomy 102, Fall 2001 2

Experimental tests of relativity

Einstein’s theories of relativity represent a rebuilding of physics from the ground up. New postulates and assumptions relate the most

basic concepts, such as the relativity of space and time and the “absoluteness” of the speed of light.

The new theory is logically consistent and mathematically very elegant.

The new theory “contains” classical physics as an approximation valid for speeds much smaller than the speed of light.

Still: it would be worthless if it didn’t agree with reality (i.e. experiments) better than classical physics.


Experimental tests of scientific theories

No scientific theory is valid unless: it is mathematically and

logically consistent, and its predictions can be

measured in experiments, and the theoretical predictions

are in precise agreement with experimental measurements.

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Galilean relativitySpecial theory of relativityMeasurements .

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Example: prediction by the “old” and “new” relativity theories for the speed of a body accelerated with constant power. Both theories are valid for low speeds but only special relativity is valid over the whole range of measurements.


Experimental tests of scientific theories (continued)

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What constitutes a valid scientific experiment? Measurements are made with

accuracy (the size of potential measurement errors) sufficient to test the predictions of prevailing theories,

and the accuracy can be estimated reliably,

and the measurements are reproducible: the same results are obtained whenever, and by whomever, the experiments are repeated.


Experimental tests of relativity (continued)

Some experiments testing special relativity: Michelson or Michelson-Morley type: speed of light

same in all directions, to extremely high accuracy. High-energy accelerators used in elementary

particle physics:• radioactive particles are seen to live much

longer when moving near light speeds than when at rest (direct observation of time dilation).

• though accelerated particles get extremely close to the speed of light, none ever exceed it.

Nuclear reactors/bombs: mass-energy equivalence (E=mc2).


Every day, many millions of special-relativity experiments

Fermi National Laboratory, near Chicago, currently the world’s highest-energy elementary-particle accelerator.


Some experiments testing general relativity: Gravitational lensing: light following gravity-induced

curves in spacetime.• Starlight deflected by the sun• Light from distant quasars deflected by

galaxies Precession of the “perihelion” of Mercury’s orbit:

matter following gravity-induced curves in spacetime. Gravitational redshifts on Earth and in the spectrum of

the sun: direct observation of gravitational time dilation.

Discovery of gravitational radiation (Nobel Prize in physics, 1993, Taylor and Hulse).

Discovery of black holes (as we’ll see).



Gravitationally-lensed quasars

“Double quasar:” mirror image of A subtracted from B leaves a faint galaxy.

Light from distant quasar Q follows warped space around galaxy G; we (at O) see images of Q in two different places on the sky. (Vertical scale greatly exaggerated.)

Images and diagram by Alan Stockton, U. Hawaii.



Results of experiments: All reproducible experiments to date have confirmed

the predictions of Einstein’s relativity theories. Few scientific theories are so well-supported by

experiment, in fact. We keep using the theory to predict new effects.

Those effects involving conditions within those for which the theory has been tested are very likely to be real. Experimental tests of these newly-predicted effects are in many cases even sterner tests of the theories.

Black holes were among the first of these “new effects” predicted by the general theory of relativity, though this was not recognized at the time.


Mid-lecture break.

Exam #1 will take place here in Hubbell Auditorium on Thursday, 4 October 2001, 2-3:15 PM. Be sure you check out the Practice Exam and its

solutions, both of which are available in the AST 102 Web Site.

The TAs will be running a Review Session, tomorrow at 8 PM (that is, after sunset), here in Hubbell Auditorium.

Bring a calculator, a writing instrument, and some experience solving problems such as have been on the homework and Practice Exam. Everything else you need will be provided, including formulas and a copy of the How Big Is That sheet.


Mid-lecture break.

Homework hints Problem 1

The terms are left loosely defined:“slightly, but noticeably” means different by 0.1% to a few % or so; “substantially” means different by a large percentage or by a factor greater than 1; “equal, to high accuracy” means exactly equal or so close that it takes an extraordinary experimental effort to tell the difference;“infinite” means infinite.


Karl Schwarzschild’s Work

Schwarzschild was interested in the physics of stars, so he solved Einstein’s field equation for the region outside a massive spherical object. In the process he discovered

some techniques, and ways of visualizing curved spacetime, that benefited others who were doing research in general relativity.• Embedding diagrams

His solution of the Einstein field equation revealed the existence of black holes. Karl Schwarzschild


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Einstein’s field equation

The field equation is the ultimate mathematical expression of Einstein’s general theory of relativity.Astronomy 102 version:“Spacetime, with its curvature, tells masses how to move; masses tell spacetime how to curve.”Physics 413 - Astronomy 554 version:


What you get when you solve the field equation

In case you’re interested (i.e. not on the exam): The solution to the field equation is a function

called the metric tensor. This function tells how much distance or time is involved for unit displacements in the spacetime coordinates.

Accordingly, the metric tensor is related to the absolute interval. Each different solution to the field equation corresponds to a different absolute interval. The absolute interval corresponding to Schwarzschild’s metric turns out to be given (in spherical coordinates) by

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r d

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Embedding diagrams, and why they’re useful

Embedding diagrams provide an analogy by which one can envision how four-dimensional spacetime can be warped, by introduction of additional, imaginary dimensions. These additional dimensions are called hyperspace.First, consider circles in flat two-dimensional space:C = 2r = d


Embedding diagrams, and why they’re useful

(continued)

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An embedding diagram.Circles in curved space still look like 2-D figures, but behave as if their centers are much further away than C/2. We can picture this as a stretching of space in the direction perpendicular to the circle.

Circle drawnin rubber sheet

Stretch!

Figure from Thorne, Black holes and time warps


Embedding diagrams, and why they’re useful (continued)

C = d if spacetime is flat(i.e. if gravity is weak)

C < d if spacetime is curved (i.e. if gravity is strong)



Schwarzschild’s view of the equatorial plane of a star

(equator)Center of star



Schwarzschild’s solution to the Einstein field equations

Results: the curvature of spacetime outside a massive star. For a given, fixed star mass M, he examined how

space and time are curved if the star is made smaller and smaller in size.

Singularity: if the star is made smaller than a certain critical size, the gravitational redshift of light (time dilation, remember) predicted by his solution was infinite! (See next slide.)


Think of this as light losing energy, as it crawls out of a deeper and deeper hole.



Implications of “Schwarzschild’s singularity”

If a star is made too small in circumference for a given mass, nothing can escape from it, not even light. • This would be a black hole, and the critical size

is the size of the black hole’s horizon. This is similar to an 18th century idea: “dark stars”

(Michell, Laplace...); that if light were subject to gravitational force, there could be stars from which light could not escape. • The critical size of Schwarzschild’s singularity

turns out to be the same as that for the 18th century dark star.

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Today in Astronomy 102: general relativity and the prediction of the existence of black holes