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The title of this thread is the title of chapter 6-3 in the Richard Feynman's book, Six Not-So-Easy Pieces (1963). (This post is related to the discussion in one other recent thread, but not to its OP.) Here is the quote (pp.125-126):

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26 minutes ago, Genady said:

The title of this thread is the title of chapter 6-3 in the Richard Feynman's book, Six Not-So-Easy Pieces (1963). (This post is related to the discussion in one other recent thread, but not to its OP.) Here is the quote (pp.125-126):

 

This could be an interesting thread.

It has a title but no point of discussion.

One question.

Do you want to discuss just 3D space or 4D spacetime or both or none of these ?

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1 hour ago, Genady said:

Any of these.

 

That didn't answer what you want to discuss about them  ?

With regard to Feynman's calculations.

I don't have access to the observational data necessary to check them. But given the pedigree of the author I would start from the premise that they are accurate.
They are the sort of observations and calculations that allow determination of intrinsic curvature in an N-dimensional manifold from within the manifold. (3 in this case).

 

Note however that in the space of the universe the curvature varies from point to point, (ie is local), according to the nearby distribution of mass.

 

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2 minutes ago, studiot said:

That didn't answer what you want to discuss about them  ?

This post actually is an elaboration belonging to the little subthread in another thread, which I simply prefer not to touch. If a moderator wants to move it there, I would not object.

Otherwise, I don't have a question regarding this topic, but many answers :)

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There is no question that, on a positively curved manifold, the circumference of a circle will be less than pi*D.
Similarily, a triangle will have angles that add up to more than 180 deg.

But this discussion originally started in another thread, where you claim the circumference of the Earth is less than pi times its diameter.
Now we both know nobody has drilled a hole through the Earth's core to measure its diameter.
The diameter is 'measured' against a co-ordinate system anchored on far-away points.
This co-ordinate system will look like Janus' link from the other thread

This co-ordinate system supplies us with x, y, z, and (-ict) co-ordinates which form the interval, S, that is the basis for the metric which determines the 'curvature' of this co-ordinate system we call space-time.
It is also the model for gravity; and this model is what is curving.

The surface of a sphere, like the Earth, is an actual physical thing, that can curve, and so, we can measure its curvature locally by comparing diameter to circumference.
Space-time, on the other hand, is simply a volume evolving in time.
It doesn't possess any property which can be curved, bent, warped, or twisted.

Incidentally, the model which is GR, can be formulated in several ways, which are equivalent.
It can be formulated in terms of varying distances and durations ( intervals ), or, in terms of the units themselves ( the rulers and clocks ) changing.
( this may be from Misner, Thorne, Wheeler Gravitation; but it's a big book, I can't find the reference right now )

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12 minutes ago, MigL said:

It can be formulated in terms of varying distances and durations ( intervals ), or, in terms of the units themselves ( the rulers and clocks ) changing.
( this may be from Misner, Thorne, Wheeler Gravitation; but it's a big book, I can't find the reference right now )

Yes, it can. Here is Thorne's popular description of this:

... The trajectory is bent around the hole; it is curved, as measured in the hole’s true, flat spacetime geometry. However, people like Einstein, who take seriously the measurements of their rubbery rulers and clocks, regard the photon as moving along a straight line through curved spacetime.

What is the real, genuine truth? Is spacetime really flat, as the above paragraphs suggest, or is it really curved? To a physicist like me this is an uninteresting question because it has no physical consequences. Both viewpoints, curved spacetime and flat, give precisely the same predictions for any measurements performed with perfect rulers and clocks, and also (it turns out) the same predictions for any measurements performed with any kind of physical apparatus whatsoever. For example, both viewpoints agree that the radial distance between the horizon and the circle in Figure 11.1, as measured by a perfect ruler, is 37 kilometers. They disagree as to whether that measured distance is the “real” distance, but such a disagreement is a matter of philosophy, not physics. Since the two viewpoints agree on the results of all experiments, they are physically equivalent. Which viewpoint tells the “real truth” is irrelevant for experiments; it is a matter for philosophers to debate, not physicists. Moreover, physicists can and do use the two viewpoints interchangeably when trying to deduce the predictions of general relativity.

Thorne, Kip. Black Holes & Time Warps: Einstein's Outrageous Legacy (pp. 400-401). 

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Like the Event Horizon of a Black Hole, I like to think of space-time as a mathematical construct, which is'superimposed' on a volume evolving in time.
This mathematical construct, or geometry, is the gravity field in GR, and is what is curving.

I maintain that neither space nor time have any property that can be curved.
And even if you could curve space like your Earth circumference/diameter analogy, we know that gravity is mostly a 'curvature' in time.
How would you show that ?

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2 hours ago, Genady said:

This post actually is an elaboration belonging to the little subthread in another thread, which I simply prefer not to touch. If a moderator wants to move it there, I would not object.

Otherwise, I don't have a question regarding this topic, but many answers :)

We don't have subthreads here, but I have put a comment into that thread you refer to since it also discusses curvature.

Meanwhile I look forward to your 'many answers' here.

Even though you don't have a question we need something specific to discuss as the topic of the thread.
This something does not have to be a question.

 

Edited by studiot
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47 minutes ago, studiot said:

... we need something specific to discuss as the topic of the thread.

This something does not have to be a question.

I will think about something for here.

I thought that maybe we can go and discuss something beyond the 3D space and 4D spacetime, namely, the internal space of particles, the bread and butter of the Standard Model. It is a difficult subject, and for simplicity we can limit ourselves to the U(1) space of EM. The main feature of it is, that it allows for a new symmetry in addition to the familiar translations in space and time, rotations, and boosts, -- gauge symmetry. One can change phases of a particle field in different spacetime points arbitrarily and if the EM potential is changed accordingly, the system behaves the same. It seems like a dirty trick, but it leads to the most precisely tested theory of all times, QED. Any ideas of "why" this gauge symmetry exists and what it means?

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And another thought. We've said, together with Thorne, that gravity can be thought as spacetime curvature or as effects on all clocks and rulers, and these two descriptions are equivalent. Yes, but...

If we want to use QFT in gravity, there is a prescription of how to do so in a curved spacetime: replace partial derivatives with covariant derivatives, stick square root of metric determinant in Lagrangian (there is one more step, don't remember now) and you got it. How to do it if gravity does not curve spacetime, but rather affects rulers and clocks?

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4 hours ago, Genady said:

I thought that maybe we can go and discuss something beyond the 3D space and 4D spacetime, namely, the internal space of particles,

Since you seem to want to discuss gravity effects in particular, I'm not sure how you want to apply these in the 'internal space of particles' or even what you mean by this.

Compared to other effects, gravitational effects are very small at short range, such as inside particles, but come into there own at galactic distances.

Are you looking for effects such as the grazing incidence of starlight passing the Sun showing up in EM radiation passing the nucleus in the otherwise empty atomic particles or  ?

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Posted (edited)
22 minutes ago, studiot said:

Since you seem to want to discuss gravity effects in particular, I'm not sure how you want to apply these in the 'internal space of particles' or even what you mean by this.

No, no, this thought has nothing to do with gravity (and I don't have any personal attraction to gravity, pan intended.) It is here only because it is about space, but this space is not affected by gravity at all. The "internal space" is a technical term in particle physics. In math it called "bundle space".

Edited by Genady
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5 hours ago, Genady said:

Any ideas of "why" this gauge symmetry exists and what it means?

It essentially just means that our description of the physics in question is over-determined, ie that there are superfluous degrees of freedom in the chosen mathematical formalism. These can be either local or global. This is why you can make specific changes to the fields without affecting the physics.

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18 minutes ago, Genady said:

No, no, this thought has nothing to do with gravity (and I don't have any personal attraction to gravity, pan intended.) It is here only because it is about space, but this space is not affected by gravity at all. The "internal space" is a technical term in particle physics. In math it called "bundle space".

Thank you for this clarification.

So we are talking about fibre bundles ?

In fact not about physical 'space' at all, but some abstract (mathematical ), space, inhabited not by physical objects but by mathematical ones.

 

2 minutes ago, Markus Hanke said:

It essentially just means that our description of the physics in question is over-determined, ie that there are superfluous degrees of freedom in the chosen mathematical formalism. These can be either local or global. This is why you can make specific changes to the fields without affecting the physics.

 

Good point, +1 

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3 minutes ago, Markus Hanke said:

It essentially just means that our description of the physics in question is over-determined, ie that there are superfluous degrees of freedom in the chosen mathematical formalism. These can be either local or global. This is why you can make specific changes to the fields without affecting the physics.

Yes, this is certainly so in the classical EM. However in QED these EM gauge transformations are coupled with local phase transformations of charged particles and this creates the mechanism of their interactions. This gauge symmetry I am asking about.

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5 minutes ago, Genady said:

Yes, this is certainly so in the classical EM. However in QED these EM gauge transformations are coupled with local phase transformations of charged particles and this creates the mechanism of their interactions. This gauge symmetry I am asking about.

Yes, I am familiar with these concepts. 

As a quick correction though, I meant to say redundant degrees of freedom, not superfluous. 

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Let me try to explain this a bit more. In a local gauge symmetry, you apply a smooth and continuous transformation to your fields at every point in spacetime, and the parameters of the transformation can vary from point to point (in global gauge symmetries, the parameters are taken to be the same everywhere). You do this by replacing the ordinary derivatives in your Lagrangian by appropriately defined gauge-covariant derivatives. For example, in QED the gauge group is U(1), so the transformation is essentially a rotation by some angle, and the corresponding gauge-covariant derivative introduces a new rank-1 object in the Lagrangian - which is just the electromagnetic vector potential, and thus the photon field.

So what does this mean? Local gauge symmetry means that fields have redundant degrees of freedom that allow a very specific type of change to happen in the field configuration at each point. Since the symmetries are continuous, by Noether’s theorem this corresponds to the existence of a conserved quantity (a charge of some kind). Consistent changes in field configuration plus conserved current equals an interaction between fields. 

So this is the central idea - redundant degrees of freedom (local gauge symmetries), consistently defined across all fields at each point on spacetime, allows for interactions between fields, and the interaction mechanism is itself a new field, as is apparent by the formalism of gauge-covariant derivatives. Without gauge symmetry, fields wouldn’t have any way to interact in this self-consistent manner. Note that in a global sense (accounting for all fields at all points in spacetime) nothing really changes at all, because all interaction currents are made up of conserved quantities - you still have the same Lagrangian after an interaction happens. All you do is ‘shift things around’, so to speak. This is the great beauty of it.

This can be very elegantly described as connection forms (called gauge potential) on fiber bundles, so a knowledgable of differential geometry is very helpful here.

Edited by Markus Hanke
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@Markus Hanke Got it. Thanks a lot!

Could you also shed light on my "another thought" (in a post way above)? Here I repeat it:

8 hours ago, Genady said:

And another thought. We've said, together with Thorne, that gravity can be thought as spacetime curvature or as effects on all clocks and rulers, and these two descriptions are equivalent. Yes, but...

If we want to use QFT in gravity, there is a prescription of how to do so in a curved spacetime: replace partial derivatives with covariant derivatives, stick square root of metric determinant in Lagrangian (there is one more step, don't remember now) and you got it. How to do it if gravity does not curve spacetime, but rather affects rulers and clocks?

 

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2 hours ago, Genady said:

Could you also shed light on my "another thought" (in a post way above)? Here I repeat it:

What exactly is meant by “effects on all clocks and rulers”, as opposed to geometry? Do you have a reference to what Kip Thorne actually stated? This doesn’t seem to make a lot of sense to me.

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2 hours ago, Markus Hanke said:

What exactly is meant by “effects on all clocks and rulers”, as opposed to geometry? Do you have a reference to what Kip Thorne actually stated? This doesn’t seem to make a lot of sense to me.

Here it is:

Isn’t it conceivable that spacetime is actually flat, but the clocks and rulers with which we measure it, and which we regard as perfect in the sense of Box 11.1, are actually rubbery? Might not even the most perfect of clocks slow down or speed up, and the most perfect of rulers shrink or expand, as we move them from point to point and change their orientations? Wouldn’t such distortions of our clocks and rulers make a truly flat spacetime appear to be curved? 

Yes. 

Figure 11.1 gives a concrete example: the measurement of circumferences and radii around a nonspinning black hole. On the left is shown an embedding diagram for the hole’s curved space. The space is curved in this diagram because we have chosen to define distances as though our rulers were not rubbery, as though they always hold their lengths fixed no matter where we place them and how we orient them. The rulers show the hole’s horizon to have a circumference of 100 kilometers A circle of twice this circumference, 200 kilometers, is drawn around the hole, and the radial distance from the horizon to that circle is measured with a perfect ruler; the result is 37 kilometers. If space were flat, that radial distance would have to be the radius of the outside circle, 200/π kilometers, minus the radius of the horizon, 100/π kilometers; that is, it would have to be 200/π – 100/π = 16 kilometers (approximately). To accommodate the radial distance’s far larger, 37-kilometer size, the surface must have the curved, trumpet-horn shape shown in the diagram.

If space is actually flat around the black hole, but our perfect rulers are rubbery and thereby fool us into thinking space is curved, then the true geometry of space must be as shown on the right in Figure 11.1, and the true distance between the horizon and the circle must be 16 kilometers, as demanded by the flat-geometry laws of Euclid. However, general relativity insists that our perfect rulers not measure this true distance. Take a ruler and lay it down circumferentially around the hole just outside the horizon (curved thick black strip with ruler markings in right part of Figure 11.1). When oriented circumferentially like this, it does measure correctly the true distance. Cut the ruler off at 37 kilometers length, as shown. It now encompasses 37 percent of the distance around the hole. Then turn the ruler so it is oriented radially (straight thick black strip with ruler markings in Figure 11.1). As it is turned, general relativity requires that it shrink. When pointed radially, its true length must have shrunk to 16 kilometers, so it will reach precisely from the horizon to the outer circle. However, the scale on its shrunken surface must claim that its length is still 37 kilometers, and therefore that the distance between horizon and circle is 37 kilometers. People like Einstein who are unaware of the ruler’s rubbery nature, and thus believe its inaccurate measurement, conclude that space is curved. However, people like you and me, who understand the rubberiness, know that the ruler has shrunk and that space is really flat. 

What could possibly make the ruler shrink, when its orientation changes? Gravity, of course. In the flat space of the right half of Figure 11.1 there resides a gravitational field that controls the sizes of fundamental particles, atomic nuclei, atoms, molecules, everything, and forces them all to shrink when laid out radially. The amount of shrinkage is great near a black hole, and smaller farther away, because the shrinkage-controlling gravitational field is generated by the hole, and its influence declines with distance. 
The shrinkage-controlling gravitational field has other effects. When a photon or any other particle flies past the hole, this field pulls on it and deflects its trajectory. The trajectory is bent around the hole; it is curved, as measured in the hole’s true, flat spacetime geometry. However, people like Einstein, who take seriously the measurements of their rubbery rulers and clocks, regard the photon as moving along a straight line through curved spacetime. 

What is the real, genuine truth? Is spacetime really flat, as the above paragraphs suggest, or is it really curved? To a physicist like me this is an uninteresting question because it has no physical consequences. Both viewpoints, curved spacetime and flat, give precisely the same predictions for any measurements performed with perfect rulers and clocks, and also (it turns out) the same predictions for any measurements performed with any kind of physical apparatus whatsoever. For example, both viewpoints agree that the radial distance between the horizon and the circle in Figure 11.1, as measured by a perfect ruler, is 37 kilometers. They disagree as to whether that measured distance is the “real” distance, but such a disagreement is a matter of philosophy, not physics. Since the two viewpoints agree on the results of all experiments, they are physically equivalent. Which viewpoint tells the “real truth” is irrelevant for experiments; it is a matter for philosophers to debate, not physicists. Moreover, physicists can and do use the two viewpoints interchangeably when trying to deduce the predictions of general relativity.

Thorne, Kip. Black Holes & Time Warps: Einstein's Outrageous Legacy (pp. 400-401).

 

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9 hours ago, Markus Hanke said:

As a quick correction though, I meant to say redundant degrees of freedom, not superfluous.

I missed that late last night.

8 hours ago, Markus Hanke said:

Let me try to explain this a bit more. In a local gauge symmetry, you apply a smooth and continuous transformation to your fields at every point in spacetime, and the parameters of the transformation can vary from point to point (in global gauge symmetries, the parameters are taken to be the same everywhere). You do this by replacing the ordinary derivatives in your Lagrangian by appropriately defined gauge-covariant derivatives. For example, in QED the gauge group is U(1), so the transformation is essentially a rotation by some angle, and the corresponding gauge-covariant derivative introduces a new rank-1 object in the Lagrangian - which is just the electromagnetic vector potential, and thus the photon field.

So what does this mean? Local gauge symmetry means that fields have redundant degrees of freedom that allow a very specific type of change to happen in the field configuration at each point. Since the symmetries are continuous, by Noether’s theorem this corresponds to the existence of a conserved quantity (a charge of some kind). Consistent changes in field configuration plus conserved current equals an interaction between fields. 

So this is the central idea - redundant degrees of freedom (local gauge symmetries), consistently defined across all fields at each point on spacetime, allows for interactions between fields, and the interaction mechanism is itself a new field, as is apparent by the formalism of gauge-covariant derivatives. Without gauge symmetry, fields wouldn’t have any way to interact in this self-consistent manner. Note that in a global sense (accounting for all fields at all points in spacetime) nothing really changes at all, because all interaction currents are made up of conserved quantities - you still have the same Lagrangian after an interaction happens. All you do is ‘shift things around’, so to speak. This is the great beauty of it.

This can be very elegantly described as connection forms (called gauge potential) on fiber bundles, so a knowledgable of differential geometry is very helpful here.

Another well thought out post thank you. +1

9 hours ago, Genady said:

Yes, this is certainly so in the classical EM. However in QED these EM gauge transformations are coupled with local phase transformations of charged particles and this creates the mechanism of their interactions. This gauge symmetry I am asking about.

Thank you both for a good discussion about current particle physics. +1

As an applied mathematician, my formal study of pp ended in 1970 and I have only followed it sporadically for interest since. My work went in other directions.

One thing that leads to many misunderstandings is the difference in terminology between Mathematics and Physics, especially for such fundamental 'objects' as Field, Vector, Tensor, Particle and Space.

It is therefore important not to mix up physical objects such as a physical fields and mathematical fields, as not all properties are interchangeable/applicable in both disciplines.

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10 minutes ago, studiot said:

It is therefore important not to mix up physical objects such as a physical fields and mathematical fields, as not all properties are interchangeable/applicable in both disciplines.

Just a little anecdote here. I was once "attacked" by somebody for saying that CMB data indicate that our universe is perhaps flat. That person decided that I'm a flat earther.

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3 minutes ago, Genady said:

Isn’t it conceivable that spacetime is actually flat, but the clocks and rulers with which we measure it, and which we regard as perfect in the sense of Box 11.1, are actually rubbery? Might not even the most perfect of clocks slow down or speed up, and the most perfect of rulers shrink or expand, as we move them from point to point and change their orientations? Wouldn’t such distortions of our clocks and rulers make a truly flat spacetime appear to be curved? 

Yes, but this is what I mean by not mixing up the measurand and the measure.

The appropriate variation should be applied to one or the other, but not both.

 

As an aside it is worth considering the origin of the word 'Frame'.

One to two centuries ago measurements were made with rigid rods and surveyors extended these by using metal chains.
There was even a unit of measure called the chain.

So it became common to imagine a coordinate system as a rigid metal framework extending in all directions from the body of interest and marked with grid marks of the  measure. This metal 'Frame' always moves along with the body, where its origin is located.

So measurements in this frame are taken against the grid marks inscribed on it. Originally frames were referred to as rigid frames.

Since the numbers are preinscribed and number is a relativistic invariant in both SR and GR, all observers will observe the same number of grid marks.

It was the genius of Einstein that led us to realise that we must therefore adjust the comparison between the grid marks on the rigid frames for different observers in some way to accomodate this, since in general each observer will see the measure between these marks as different from the ones in his own rigid frame.

This matchng can be done in one of two ways. Either by transformation (SR) or by applying a 'metric' (GR)

 

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