# Gravity and light

By Daniel Holz | July 22, 2009 12:15 am

A few hours ago the longest total solar eclipse of the Century swept across Asia. And a few days ago Evalyn Gates provided a wonderful guest post on gravitational lensing. This seems like an opportune time to note that gravitational lensing and total solar eclipses are inextricably linked.

One of the most interesting predictions of Einstein’s new theory of relativity was that gravity would cause light to bend. Imagine you are looking at a distant source of light, for example a star, or a faraway galaxy, or a quasar at the edge of the Universe. And let’s assume that, along the line-of-sight to the distant source there’s a massive object, for example the Sun, or a black hole, or a galaxy, or a cluster of galaxies. The gravity from the massive object will “pull” on the photons as they pass, shifting their paths, and thereby affecting the image that we see in our telescopes. In the simple case of a distant point source of light (e.g., a far away star), and a compact spherically symmetric lens (e.g., a black hole), the bending angle is given by
\$latex displaystyle theta=(G/c^2)4M/r\$
In this equation M is the mass of the lens, r is the minimum distance between the (unperturbed) line-of-sight to the source and the lens, G is the gravitational constant, and c is the speed of light. This was a crucial prediction of Einstein’s new theory, and one way to test it was to see if the stars on the sky “jump” as the Sun (which is quite massive, and traverses the sky quite briskly) comes nearby on the sky. If you plug in the appropriate numbers above ((G/c^2)*M_sun = 1.5 km [geometric units], R_sun = 700,000 km), you find that a star should shift on the sky by 1.75 arcseconds (8.57e-6 radians) as the Sun approaches. There’s one slight snag in measuring this effect: the Sun is sort of bright. When it’s up in the sky it can be a little hard to see what the stars are doing. By the time it’s dark and you can see stars, the Sun is far away on the sky (e.g., below the horizon), and there’s no longer a measurable effect. But nature conveniently provides a very elegant solution to this problem: the total Solar eclipse. In one of the more mysterious coincidences (or is it an argument for “intelligent design”?), it turns out that the Moon and the Sun have very similar angular sizes, when seen from Earth. So every now and then the Moon crosses right in front of the Sun and blocks it out. The sky goes dark. The stars come out in the middle of the day. It even becomes possible to see stars near the very edge of the Sun. Nature conveniently provides the perfect system in which to validate the general relativity prediction of gravitational lensing.

We have a habit in science of simplifying the historical progression. Einstein’s initial 1911 prediction was off by a factor of two (giving hope to us mere mortals). Over the next few years a number of expeditions were mounted to test his prediction, but all of them failed (e.g., bad weather, World War I). This gave Einstein time to discover his error, and in 1915 he fixed his result, arriving at the equation above. The definitive (though subsequently controversial) measurement was performed by Sir Arthur Eddington in 1919. He observed the positions of stars during a total eclipse, claimed to confirm Einstein’s prediction, and vaulted Einstein to fame. In one of the best newspaper headlines ever, the New York Times front page page 17 announced: “LIGHTS ALL ASKEW IN THE HEAVENS; Men of Science More or Less Agog Over Results of Eclipse Observations”.

We’ve come a long way. Gravitational lensing is now one of our best probes of the Universe, revealing the presence of dark matter, and maybe eventually becoming a sensitive probe of dark energy. I’m super bummed I didn’t get to see the total eclipse a few hours ago. But I have every confidence that the stars were all appropriately askew, and that people were appropriately agog.

CATEGORIZED UNDER: Science
• http://airminded.org Brett

It’s Sir Arthur (or Sir Arthur Eddington, which would be better here), never, ever Sir Eddington. Sorry to be pedantic, it’s a pet hate of mine.

• RichardW

But used to brilliant comic effect in “The Sopranos”, with Ben Kingsley being addressed as “Sir Kingsley”. ðŸ˜‰

• James

“One of the most interesting predictions of Einsteinâ€™s new theory of relativity was that gravity would cause light to bend.”

I think it is worth mentioning that the bending of light due to gravity was NOT a prediction of general relativity.

As early as 1704 in his Opticks, Newton predicted the effect. However, the speed of light was not known a the time (or even whether it was finite) so no quantitative prediction could be made. This was rectified by the end of the 18th century and the Newtonian calculation could be made, though experimental limitations forbade any test at the time.

In 1911 Einstein applied his early ideas of relativistic gravity to the problem and got the same answer as the Newtonian model. In 1915, when his theory was approaching completion, he realised the earlier calculation was wrong, and the deviation of light should be twice the Newtonian value. This has since been confined numerous times.

So, it was the degree of the bending of light – not the idea of the bending itself – that was new in general relativity.

• http://lablemminglounge.blogspot.com/ Lab Lemming

“it turns out that the Moon and the Sun have very similar angular sizes, when seen from Earth.”

It is worth pointing out that this is just a temporary coincidence, as the moon is getting farther away due to tidal dissipation, and the Sun is growing slowly brighter.

• Jorge Pullin

It was page actually on page 17 in the New York Times, right next to news about the Bosnian conflict.

• http://www.myreferenceframe.com/ My Reference Frame

Who cares about exactly how British titles of nobility are written?? Anyway, it would be interesting to see the results of any experiments done during this event re this question.

• ND

Have there been repeats of Eddington’s observation at other solar eclipses?

• http://www.teamsikorski.com Spiv

So, for r you’re using radius to the edge of the photosphere?

My first thought was “Doesn’t the moon’s mass come in to play too?” followed by “why not just use a new moon?”

Well, I did the math. For anyone wondering it’s about 1.36*10^-5 arcseconds of shift. That would be a telescope with the resolving power equivalent to over 5 miles of diameter to discern it.

Was fun, though. Thanks for the lead on an interesting physics/math problem.

• http://danielholz.com daniel

@Brett. Fair enough. Sir Arthur just didn’t sound right (King Arthur? Arthur Conan Doyle?). I’ve put in his full name.

@Jorge. Thanks for pointing that out! I’ve updated the post. For years now I’ve had a clear image of that front page in my head, with the “lights askew” article in the bottom right corner. I guess I’ve been hallucinating–wanting to believe GR had a more glamorous debut.

@ND. There have been many subsequent measurements, some of them detailed here. In particular, radio observations have definitively confirmed the predictions.

@Spiv. As your number indicates, the Sun is pretty damn massive compared to the Earth or Moon. Or even Jupiter, for that matter.

• http://danielholz.com daniel

@James. Newtonian gravity does indeed ‘predict’ light bending, but as you point out, gets the factor of two wrong. Regardless, the main point here was that this was a way to ‘falsify’ Einstein’s newfangled theory. The bending of light was essential and fundamental to the theory. Had the bending of light not been observed, people would have thrown out Einstein, not Newton. Einstein’s quote on the issue: (Had the light bending not been observed to agree with my prediction), “Then I would feel sorry for the good Lord. The theory is correct anyway.”

• James

Well, being British I felt I had to do my duty and pedantically stick up for the Brit who got the idea first ðŸ˜‰

By the way, if the Newtonian result had turned out to be correct, presumably gravitational lensing would be still an observable (and useful) phenomenon?

• James

Hence, relativity isn’t really important here – it just contrubutes a scale factor of 2 (playing devil’s advocate…)

• http://www.digihara.com John

Did Einstein actually make a factor 2 mistake initially, or was the GR that he was
considering at that time incorrect? He had a number of GR theories– one in 1913, one in 1914 ect. until he got the one proposed by David Hilbert.

• Shantanu

Note that because of the same reason why GR bending is twice Newtonian value, protons
which travel in particle accelerators at the speed of light experience a net acceleration twice
the acceleration due to gravity (~ 20 m/sec^2). See this interesting
colloquium at Fermilab which discusses this and other interesting tests of GR at Tevatron.

• Pentcho Valev

In 1911 Einstein said that the speed of light varied with the gravitational potential V in accordance with the equation c’=c(1+V/c^2), which was in fact a prediction of Newton’s emission theory of light. In 1915 he added a factor of two and the equation became c’=c(1+2V/c^2). The problem was (and still is) that the 1911 equation is consistent, and the 1915 equation INCONSISTENT, with the gravitational redshift factor 1+V/c^2 experimentally confirmed by Pound and Rebka.

Pentcho Valev
pvalev@yahoo.com

• Jimbo

John,
Yes, it was AE’s mistake, as his early version was not the same as the final field eqs. Its the factor of 2 that was missing. Daniel chose to obfuscate the deflection via a messy formula, whereas its crystal clear if one just writes it in dimensionless form,
theta = 2Rs/r, where Rs is the Schwarzschild radius of the sun, ~ 3Km.

• http://telescoper.wordpress.com Peter Coles

The reason for the factor of two error is that Einstein’s earlier calculation didn’t include spatial curvature (which is essentially the new ingredient of general relativity). The time-time components give you the same answer for the deflection as Newtonian theory; adding the space-space components (which he did in 1915) gives you an extra deflection which is the same size as the original, hence the factor of two. Putting this another way, using energy/mass alone gives you the Newtonian value but in GR momentum also gravitates, giving you the factor two (for highly relativistic particles).

I should also mention that you just missed the 90th anniversary of the Eddington experiment by a couple of months…

• http://telescoper.wordpress.com Peter Coles

PS. In the Opticks , Newton wrote

Do not Bodies act upon Light at a distance, and by their action bend its Rays; and is not this action … strongest at the least distance?

I’ve certainly always interpreted this as meaning that he thought light would be deflected by gravity just like matter would be, i.e. according to the inverse-square law. However, many historians of science disagree with this interpretation and think he had in mind something rather more like some sort of gravitational refraction effect. In any case no calculations of Newton’s on this topic survive so I think we’ll probably never know.

The first published calculation of the “Newtonian” gravitational deflection of light that I’ve ever found was by Johann Georg von Soldner, and appeared in 1801.

• http://www.jumplive.com/tech.html chimpanzee

I observed the eclipse in Eastern Tibetan Plateau @15,000 ft, with Gongga Shan (22,800 ft Everest like peak) in foreground:

http://www.eclipse-chaser.com/2009/index.html

I measured a light & temperature profile from 1st to 4th contact. I met a Polish group observing nearby, including an atomic scientist (Inst of Atomic Energy/Radiation Protection Measurements Lab/Otwock-Swierk, Poland)

The logistics of solar eclipse expedition (like Eddington in 1919) cannot be underestimated. Lots of equipment, & stress over weather. Seth Shostak/SETI & Glenn Schneider/U. of Arizona/Steward Observatory (chief instrument scientist for Hubble’s WFPAC2) observed from Wuhan (which was compromised by clouds), & Jay Pasachoff (Harvard alumni, on sabbatical @Caltech) observed near Shanghai (also compromised by clouds). Many emails were exchanged amongst us (incl Dan Fischer, German science writer), trying to dodge the China monsoonal flow. In the end, I followed the move of Quanzhi Ye (meteorlogical student @Sun Yat Sen Univ) who went to Shangri-La (100km west of Daocheng), in Eastern Tibetan Plateau. I ended up at his original site (which he abandoned), Zimei Pass @15,000 ft.

A friend of mine (who knows a Silicon Valley IPO) flew a private jet to Marshall Islands (site of Hydrogen Bomb testing) & got clear skies.

• Crackpot#1

We have, as you all might know, according Harvard’s Christopher Stubbs that “Understanding dark energy is arguably the most profound problem in contemporary physics”. I can not help but point out the importance of the connection of GRAVITY and LIGHT that is not exactly mentioned in the physics textbook or in this blog.

I can place a .21 Kg test mass underneath a cold source. When the temperature of the test mass increases to ~400 C, its gravitational mass increases by 22%. This is a 47 gm gain in gravitational mass, which according to the hollowed principle of equivalence, is thought to be equal to the inert or inertial mass .

The are table top experiments in the past that have required a paradigm shift like the photoelectric effect, the black body spectrum and Rutherford’s gold foil experiment.

It took 15 years to get the above results which runs contrary to Count Rumford study of the relationship between heat (light) and gravity. Four other similar results and a radiation-based gravity theory to go with them can be found by googling viXra: 0907.0018.

• http://cosmicdarkmatter.com Tissa Perera

According to my long range gravity concept, the light bending formula is is as follows:

deflection angle of light = 2MG/c^2/r + 2MG/c^2/R

where R = 3Kpc a constant. and when r > R.
Therefore the light always deflect at a constant angle 2MG/c^2/R when r>>>R.
Look up:
http://cosmicdarkmatter.com/Newtonian_Dynamics.html

See if anyone can veryfy that fact and prove me right if not wrong.

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