# Does this thought experiment show that Space must be curved?

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• peterraymond
In summary, this demonstration demonstrates how light is deflected by the gravitational force experienced in any small region of space-time. This is evidence that spacetime (not space!) is curved. However, this proof is not conclusive, as we need something else in order to demonstrate that spacetime is curved.
peterraymond
TL;DR Summary
A mental experiment that shows that gravity bends light
I'm hoping this is basic and obvious, but assume it's not to the general public.

Ignore quantum mechanics and diffraction and assume a gun that can fire photons that each hit the center of a remote target. Place one of these and a conventional gun that shoots bullets at 1000 m/sec inside of a black box. Set the target range for your photon gun at 3000 km and for your conventional gun at 10 m, so travel time for each is 10 msec.

In a box where an observer feels weightless align the guns to their targets. No particular positioning in space is needed and you must still get direct hits if the box is moving in any way where the observer feels weightless throughout the box, because each condition is identical to the observer. Shots will still stay on target if for instance the box is accelerating at 100 m/sec^2 as it falls straight towards a far off super massive black hole.

If at the moment the shots are fired rocket motors start accelerating the targets "upward" at 10 m/s^2, over 10 msec each target will have moved 0.5mm. Or, the whole box could suddenly start accelerating at 10 m/sec^2, or that acceleration could be continuous. In all these cases the observer inside the box would see a miss at both targets because acceleration has moved the targets out of position during the time of flight. Direct measurements from the accelerating walls of the box though would make both paths appear curved.

Identical for the observer is having the box sit "stationary" on the surface of a large body where acceleration due to gravity is 10 m/sec^2. The observer can't tell the difference and the photon and the bullet must both still miss their targets by 0.5mm. Both appear to "drop" 0.5 mm over 10 msec. In this case though, Newtonian mechanics would say that the bullet path drops because the bullet is attracted to the mass of the large body. For the photon there is no mass, no gravitational force and no Newtonian way to calculate the apparent curve in its path. Both paths would be measured to curve, but only because this space was curved by gravity. If this were not true, the observer would need to somehow see through the walls of the box and some measures of motion would be absolute, not relative.

Yes?

That's really just a demonstration of the Equivalence Principle.

"the gravitational force experienced in any small region of space-time is the same as the pseudo-force experienced by an observer in an accelerated frame of reference."

peterraymond said:
Ignore quantum mechanics and diffraction and assume a gun that can fire photons…
(Trying to ignore quantum mechanics in a discussion involving photon behavior is like trying to ignore water in a discussion of ocean waves. But we know that when you say “photon” you really mean “flash of light” and we’re making the correction for you as we read)

And with that said….
Thought experiments of the sort that you describe do show that light should be deflected by gravity. Googling for “equivalence principle deflection of light” will find a number of similar thought experiments, some a bit easier to state and understand than others.

However, we can’t get from there to proof that spacetime (not space!) is curved. For that we need something where the deflection from spacetime curvature is different than the deflection predicted by the Newtonian model, such as when light passes by a gravitating body. Eddington’s observations of starlight during the 1919 solar eclipse might be the first such test.

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If spacetime was curved enough, we would start to see visual distortions analogous to curved mirrors in the carnival funhouse.

Some people point out that the art of Escher illustrates some kinds of distortions.

We haven't seen such distortions. But if we did, it would provide strong evidence for the existence and the sign of curvature.

peterraymond said:
In this case though, Newtonian mechanics would say that the bullet path drops because the bullet is attracted to the mass of the large body.
Yes.

peterraymond said:
For the photon there is no mass, no gravitational force and no Newtonian way to calculate the apparent curve in its path.
Yes. This is one respect in which Newtonian mechanics does not really obey the equivalence principle, since in the rocket accelerating in free space Newtonian mechanics does predict that the light ray's path will appear to curve in the rocket's frame.

peterraymond said:
Both paths would be measured to curve, but only because this space was curved by gravity.
I'm unclear which theory you think this claim applies to. Do you think it applies to Newtonian mechanics? To GR?

A thought experiment can never prove anything.

peterraymond said:
Summary:: A mental experiment that shows that gravity bends light

Yes?
As others have said, this is a use of the equivalence principle. However, although all metric theories of gravity (any curved spacetime model) respect the equivalence principle necessarily, so can other theories. For example, Newtonian gravity can be argued to do so. Since ##F=ma## and ##F=GMm/r^2## then the acceleration due to gravity is ##a=GM/r^2##. Since that depends only on position (notably, it doesn't depend on mass so it's not immediately implausible that light would be affected too) Newtonian gravity respects the equivalence principle too.

An experiment that showed that gravity did not respect the equivalence principle would show that GR (and any other metric theory) could not be correct. But the converse is not true.

Nugatory said:
(Trying to ignore quantum mechanics in a discussion involving photon behavior is like trying to ignore water in a discussion of ocean waves. But we know that when you say “photon” you really mean “flash of light” and we’re making the correction for you as we read)

Thank you for making that correction. I made a judgement that I did not want to talk about and get bogged down by: photons are particles and waves, an optical system has a diffraction limited spot size and similar stuff that seemed off topic to me. I could instead have talked of a collimated beam and measuring its center of energy, or talk about imagining a photon traveling down the center of that beam, but I was trying to keep the post as short as I could. Yes I realize, it should be as short as possible, but no shorter.

However, we can’t get from there to proof that spacetime (not space!) is curved. For that we need something where the deflection from spacetime curvature is different than the deflection predicted by the Newtonian model, such as when light passes by a gravitating body. Eddington’s observations of starlight during the 1919 solar eclipse might be the first such test.

I assume that the path of photons would curve in this experiment in the presence of gravity, but that Newtonian mechanics would predict no deflection. My experiment though clearly has the problem that it would be virtually impossible to conduct. Unlike Eddington's observations.

peterraymond said:
I assume that the path of photons….
Photons aren’t what you think they are. Whether you know it or not, you are thinking in terms of a light beam or a flash of light or similar, not photons.
…would curve in this experiment in the presence of gravity, but that Newtonian mechanics would predict no deflection.
As @Ibix points out in #7 above, it is by no means clear that Newtonian mechanics does not predict deflection (and note that Newtonian mechanics does not require that light be massless).

That’s why the important question is not whether light deflects, but by how much. The GR prediction agrees with Newton for a uniform gravitational field and Einstein’s accelerating elevator (no gravity in the latter case, but the equivalence principle says the effect should be same) but not for the case of a light beam passing near a gravitating mass.

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I realized before I wrote that my photon gun is impossible and tried to indicate that. You can determine statistically the probability of a photon landing in different places, but you can't contrain it to land in any particular one. You could in theory use a large telescope to send a large diameter collimated beam and then use a detector array larger than the resulting diffraction limited spot to measures its translation. Or more practically you could have a smaller detector and use a lens to focus a portion of the beam to a diffration limited spot. Shift in the focused spot would then be a measure of the change in angle of the beam. The whole experiment is impossible though in many other ways. How do you align optics sitting in a gravitational field? Not optically of course. And how do you accelerate with rocket motors a black box that is 3000 km long and keep everything aligned?

The experiment does assume knowledge that photons have no mass so that in an uncurved Newtonian universe the vertical acceleration of photons due to gravity would be zero divided by zero.

I guess I was just intrigued by the idea that in a gravitational field, like bullets shot horizontally, photons drop like a rock and thought that this idea might make curvature of light by gravity one step clearer to a general audience.

One of the things I'm afraid I don't understand is how this measured deflection of photons in a gravitational field is different from the bending of light passing gravitational bodies, except for that fact that this experiment is impossible to conduct, unlike real experiments with useful results.

peterraymond said:
One of the things I'm afraid I don't understand is how this measured deflection of photons in a gravitational field is different from the bending of light passing gravitational bodies, except for that fact that this experiment is impossible to conduct, unlike real experiments with useful results.
One case has a homogeneous gravitational field and the other does not.

peterraymond said:
how this measured deflection of photons in a gravitational field is different from the bending of light passing gravitational bodies
If by "gravitational field" you mean the "field" in an accelerating box in flat spacetime, the obvious difference is that spacetime is flat in that case, whereas in the case of bending of light passing gravitational bodies spacetime is curved. Locally these two cases are indistinguishable, but over large enough scales that tidal gravity (i.e., spacetime curvature) is detectable of course they are not.

peterraymond said:
I guess I was just intrigued by the idea that in a gravitational field, like bullets shot horizontally, photons drop like a rock and thought that this idea might make curvature of light by gravity one step clearer to a general audience.
The point is not whether light is deflected by a large mass, but by how much? The calculation in Newtonian mechanics is simple: we either assume light is not influenced by gravity or that it has the common gravitational acceleration that massive bodies do.

The explanation in GR is likewise simple: it's the equivalence principle.

The difficult part is the calculation of a specific deflection around the Sun for the purpose of experimental verification of GR. It turns out this is approximately twice the Newtonian estimate - for this particular case. See, for example, Chapter 9 of Hartle's Introduction to GR.

There is a thread on this here:

https://www.physicsforums.com/threa...ht-is-twice-the-Newtonian-prediction.1003623/

I did go to:

https://www.physicsforums.com/threa...ht-is-twice-the-Newtonian-prediction.1003623/

If you want relatively simple facts you can find them on the internet. If you want to understand something more complicated you need a book. That's what I need.

I believe I might be thinking of the curvature of space, not space-time. I've empirically designed a flex lead - which is a catenary - for a magnetic disk drive in MatLab. I imagine doing something similar for light passing a star. I would work forwards and back from the closest approach to the star where the light path and gravity are perpendicular. I'd then use local gravity to directly calculate a local curvature for the path similar to what I did in my original thought experiment. After some chosen path length I'd calculate a new end position and angle. As you move away from the star gravity has one component perpendicular to the path and one parallel. I think my error is that I'd just use the perpendicular components to calculate new curvatures and ignore the parallel components.

I appreciate every comment and I'll read any further ones, but it's time to pick a book. There was a free online one suggested in the other thread.

peterraymond said:
I believe I might be thinking of the curvature of space, not space-time.
The scenario in your OP has nothing to do with either spacetime curvature or space curvature. It has to do with proper acceleration and the equivalence principle. But the whole point of the equivalence principle is that if you limit yourself to just local observations (what you can observe inside the box in your OP), you can't tell whether the spacetime you're in is flat or curved.

The fact that Newtonian mechanics (arguably) does not predict that a photon's path would appear curved in the accelerated box sitting on the surface of the Earth just shows that Newtonian mechanics (arguably) does not fully respect the equivalence principle. Again, this has nothing to do with either spacetime curvature or space curvature.

peterraymond said:
I would work forwards and back from the closest approach to the star where the light path and gravity are perpendicular. I'd then use local gravity to directly calculate a local curvature for the path similar to what I did in my original thought experiment. After some chosen path length I'd calculate a new end position and angle.
The process you describe here is somewhat similar to one way of computing the global light bending by a gravitating mass like the Sun. However, there are several key differences in the correct computation as it would be done in GR (and this computation is given as an exercise in many GR textbooks):

First, the path of the light ray is a path in spacetime, not space, and it is not a curved path in spacetime: it is a geodesic, with zero path curvature. Its projection into a particular spacelike surface of constant time "looks" curved because of the curvature of spacetime.

Second, the local contribution to "bending" at a given point on the light ray's worldline that you are thinking of is assuming an accelerated observer who is "hovering" at just the right fixed altitude to see the light ray fly past him, through his local "box", and appear to "fall" downward as it passes through the "box".

Third, in order to correctly add up the local contributions to "bending" all along the light ray's worldline, you need to take spacetime curvature into account. If you don't, you will get the "Newtonian" answer, which is half of the correct (GR) answer. To put it another way, spacetime curvature means that the local "frames" of the accelerated observers all along the light ray's worldline do not "line up" globally the way they would in flat spacetime (or in Newtonian gravity).

It is true that, because the particular scenario being considered (bending of light by a single, spherically symmetric gravitating body) is stationary (which basically means we can view it as a stack of 3-dimensional "spaces" that are all the same--"space" does not change with "time"), you can project the light ray's worldline into a single "space" (a spacelike 3-surface of constant time) and if you squint at the computation just right in this projection, it can seem like it is "space curvature" of the 3-dimensional space that is causing the bending to be twice as much as the "Newtonian" value. You will find plenty of pop science sources, and even some textbooks, that explain it this way. But this explanation, while not exactly incorrect, is IMO misleading because it makes use of a very specific feature of this particular scenario (that it is stationary) and does not generalize to other scenarios. Whereas the general method of computing the null geodesic that the light ray follows in curved spacetime works for any scenario.

Note also that the general method I just referred to does not require you to imagine a family of accelerated "hovering" observers all along the light ray's worldline. That particular method also only works in a stationary spacetime. But there are more general methods of computing geodesics that work in any spacetime.

peterraymond said:
As you move away from the star gravity has one component perpendicular to the path and one parallel. I think my error is that I'd just use the perpendicular components to calculate new curvatures and ignore the parallel components.
I'm not sure this corresponds to anything in the actual math. See above for what I think is a better description.

PeterDonis said:
The fact that Newtonian mechanics (arguably) does not predict that a photon's path would appear curved in the accelerated box sitting on the surface of the Earth just shows that Newtonian mechanics (arguably) does not fully respect the equivalence principle. Again, this has nothing to do with either spacetime curvature or space curvature.

I'm only on chapter 3 in the reading and it supports the idea that you need to understand the foundations and proceed step by step from there. Otherwise it's like trying to skip addition and go straight to multiplication.

Having admitted that and using a term I certainly don't fully understand, it seems like both bullet and photon follow straight lines through curved space-time and GR explains why equivalence is not violated by the measured paths in the box. Also, maybe this is just semantics.

Just as an aside, maybe the experiment should have had two telescopes at the ends of the box both looking at a single far off star, instead of the imaginary photon gun. I calculate an apparent shift of 69 micro-arc-sec between the two telescopes for a local observer.

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peterraymond said:
it seems like both bullet and photon follow straight lines through curved space-time
That is correct. They are different straight lines (more precisely, geodesics) through curved spacetime, but they are both straight.

peterraymond said:
maybe the experiment should have had two telescopes at the ends of the box both looking at a single far off star, instead of the imaginary photon gun. I calculate an apparent shift of 69 micro-arc-sec between the two telescopes for a local observer.
How are you calculating this?

vanhees71

## 1. What is a thought experiment?

A thought experiment is a mental exercise used to explore a scientific or philosophical concept. It involves using imagination and logical reasoning to understand a complex idea or phenomenon.

## 2. How does a thought experiment relate to the concept of curved space?

A thought experiment can be used to help visualize and understand the concept of curved space. By using imagination and logical reasoning, one can explore the implications and consequences of a curved space without actually physically experiencing it.

## 3. What is meant by "curved space"?

In physics, space is described as being curved when the geometry of space is not flat, but rather distorted or curved due to the presence of massive objects. This is a fundamental concept in Einstein's theory of general relativity.

## 4. How can a thought experiment demonstrate that space must be curved?

A thought experiment can demonstrate that space must be curved by exploring the consequences of a curved space, such as the bending of light or the effects on the motion of objects. By using logical reasoning and mathematical calculations, one can show that these consequences can only occur in a curved space.

## 5. Are there any limitations to using thought experiments to understand the concept of curved space?

While thought experiments can provide valuable insights into the concept of curved space, they are limited by the fact that they are not based on direct observation or experimentation. Therefore, they may not accurately reflect the true nature of space, and further research and experimentation are necessary to fully understand the concept.

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