Understanding Einstein Field Equations Through Tensor Calculus

[Passionflower]

That sounds vaguely quantifiable. Where's this come from?

JDoolin has got a point.

Peter, could you give us the formula on which you base this?

 

[PeterDonis]

That sounds vaguely quantifiable. Where's this come from?

It seems to be a fairly common observation in relativity textbooks; it's based on the fact that a "naive" Newtonian calculation of the bending of light by the Sun gives an answer that's half the general relativistic value. I don't have time right now to dig up "official" references, but this page:

http://www.mathpages.com/rr/s6-03/6-03.htm

has a pretty good, if somewhat brief, discussion.

Edit: I should also note that I was using language loosely in referring to the "Newtonian" part of the bending of light as "space curvature" and the rest as "time curvature". The full general relativistic deflection is what will show up in any measurements, even if they are "purely spatial" in nature. For example, suppose we sent two satellites out and positioned them in such a way that they could send out laser pulses to Earth that would appear to come from opposite sides of the Sun and just grazing the Sun's surface. If we then measured the three angles of the triangle formed by the two pulses and a third pulse sent between the satellites at the same time as the first two pulses were emitted (we assume that this third pulse's path is far enough away from the Sun that any relativistic effects on the path due to the Sun's gravity are unmeasurable), we would find that the sum of the angles was larger than 180 degrees by the full GR deflection amount, summed for both laser pulses.

 

[Passionflower]

Edit: I should also note that I was using language loosely in referring to the "Newtonian" part of the bending of light as "space curvature" and the rest as "time curvature". The full general relativistic deflection is what will show up in any measurements, even if they are "purely spatial" in nature. For example, suppose we sent two satellites out and positioned them in such a way that they could send out laser pulses to Earth that would appear to come from opposite sides of the Sun and just grazing the Sun's surface. If we then measured the three angles of the triangle formed by the two pulses and a third pulse sent between the satellites at the same time as the first two pulses were emitted (we assume that this third pulse's path is far enough away from the Sun that any relativistic effects on the path due to the Sun's gravity are unmeasurable), we would find that the sum of the angles was larger than 180 degrees by the full GR deflection amount, summed for both laser pulses.

So are you no longer saying that the space and time curvature is each exactly 50% in this scenario? If you are still saying that would it be fair for me to ask you to back that up with math and show where exactly we have 50% spatial and 50% temporal curvature?

 

[PeterDonis]

So are you no longer saying that the space and time curvature is each exactly 50% in this scenario?

My comment about loose use of language was basically retracting that statement, yes, but it should be noted that that does not mean there are no timing effects involved in the scenario. The laser pulses passing close to the Sun would be time delayed (the "Shapiro time delay") as well as arriving at Earth at a slightly increased angle.

Edit: I should also note that I remember reading in at least some sources (as I noted in an earlier post, I haven't had time to check references for specific quotes) that the existence of the Shapiro time delay is the reason for attributing the fact that the GR result for the angle of light bending is twice the Newtonian one to the presence of "time curvature" as well as "space curvature" (hence the 50-50 split). However, as I said, I agree that's a loose use of language.

 

[PeterDonis]

...the existence of the Shapiro time delay is the reason for attributing the fact that the GR result for the angle of light bending is twice the Newtonian one to the presence of "time curvature" as well as "space curvature" (hence the 50-50 split).

Hmm...I may have been remembering it backwards. In going back through the huge collection of bookmarks in my browser, I noticed a link to the Gravity Probe B site:

http://einstein.stanford.edu/SPACETIME/spacetime3.html

This has the following quote in the section on light bending by the Sun:

If Mercury's perihelion shift led to the acceptance of general relativity among Einstein's peers, then light deflection made him famous with the public. He had already found in 1911 that the equivalence principle implies some light deflection, since a beam of light sent horizontally across a room will appear to bend toward the floor if the room is accelerating upwards. (Similar arguments had in fact been proposed on purely Newtonian grounds by Henry Cavendish in 1784 and Johann Georg von Soldner in 1803.) In 1915, however, Einstein realized that space curvature doubles the size of the effect, and that it might be possible to detect it by observing the bending of light from background stars around the sun during a solar eclipse.

So in this interpretation, the doubling of the light-bending effect is due to the incorporation of *space* curvature in GR.

 

[Passionflower]

So in this interpretation, the doubling of the light-bending effect is due to the incorporation of *space* curvature in GR.

Yes that is often said but forgive me for not taking this as gospel. I am not claiming it is wrong but I rather see formulas that actually show that. You think that is unreasonable?

For starters in light bending calculations the r coordinate is actually used as a physical distance so spatially speaking there is not much going on as space is now assumed spatially flat (e.g. rho is assumed equal to r). On the other hand light actually decelerates when attracted by a gravitational field and my intuition would say that is the reason for the 'double whammy' not spatial curvature.

Surely I must be wrong, for my understanding is little, and greater minds on this website will obviously have no trouble at all to demonstrate I am wrong and lack understanding by using the appropriate formulas. So I remain.

 

[PeterDonis]

Yes that is often said but forgive me for not taking this as gospel. I am not claiming it is wrong but I rather see formulas that actually show that.

I'm not sure I take it as gospel either. I can see the general line of argument: the Newtonian calculation gives an answer D for the deflection, but in Newtonian physics space is assumed to be flat; GR allows space to be curved and gets the answer 2D for the deflection. However, I don't know that there's an equation that cleanly shows the GR deflection as a sum of two terms, one for "space curvature" and one for "Newtonian effect". In the derivation on the page I linked to earlier, the factor of 2 in GR comes in as a multiplier, not a separate term added to the Newtonian one. (Also, it's a matter of opinion whether the Newtonian effect is aptly described as "time curvature"; the Gravity Probe B quote I gave notes that Einstein's original argument for light bending was based on the equivalence principle, which is supposed to hold in local inertial frames that look like small pieces of *flat* spacetime.)

 

[PeterDonis]

So I would guess that you are correct that only a static metric can be spatially flat in the (B) sense.

On further thinking, I'm no longer sure this is right, but I could easily be misunderstanding something. I've started a new thread about this question here:

https://www.physicsforums.com/showthread.php?t=446589