IT pro, part-time physicist

Space is slightly curved by our Sun's mass, explaining the precession of planets and bending of light. How does this non-time-dependent tiny curvature of space relate to the big curvature of space-time needed for elliptical planetary orbits?

I take it that this question is inspired by often-seen illustrations of space, as a rubber sheet, being bent by the big and heavy Sun in the center.

Please, forget you ever saw this illustration. I stopped using it in explanations years ago, because it is, to say the least, grossly misleading.

While it is indeed true that the Sun introduces a tiny amount of spatial curvature, the gravity that you feel, Newtonian gravity that is, is due almost entirely to time curvature. Spatial curvature only introduces a correction that amounts to a few parts in a billion. Up until the second half of the 20th century, the very small, anomalous perihelion advance of Mercury was the only observable effect of spatial curvature.

In other words, what you and I experience as gravity has to do with the (slight) differences at the rate at which otherwise identical clocks tick, depending on their distance from the source of gravity.

Orbits are what they are because planets move not just in space but also in time. They follow what would be a so-called geodesic in spacetime. In the absence of gravity, these geodesics would be straight lines. The presence of gravity bends geodesics towards the source of gravity, basically in the direction where clocks tick more slowly. Imagine a planet that is initially standing still. So it is moving forward in time, but it is not moving in space. But its forward-in-time geodesic motion is bent slowly towards the Sun. From our perspective (keeping in mind that we, too, move forward in time along with the planet) this appears as though the planet started to accelerate towards the Sun. Now planets do not begin their existence standing still; they are moving “sideways”, in addition to moving forward in time. So the bending of their spacetime trajectory towards the Sun manifests itself as a curved orbit around the Sun.

This does not mean that space is curved. It simply means that the same planet, e.g., the Earth, that is moving “sideways” at 30 km/s is also moving forward in time at the equivalent of nearly 300,000 km/s, and it will be the “time curvature” of spacetime that will dominate how its trajectory changes in the presence of the Sun.

When physicists say that a particle is really an "excitation in a field" does this refer to the particle before or after the wave-function collapse? Or both?

Wavefunction collapse has nothing to do with it. Wavefunction collapse describes the act of observation: it refers to the fact that when a specific property of a quantum system is observed, it has a numerical value, but at all other times, it is “operator valued”, meaning a mathematical representation of a quantity that does not obey conventional rules of arithmetic.

However, while conventional quantum particle mechanics can describe how a particle moves about and the probability densities that characterize the outcome of various measurements, it cannot to one of the most basic of things that we observe in nature: It cannot tell us how, e.g., a photon is created or absorbed. It can only tell us what the photon does in between the two events.

Quantum field theory offers a completely different take. Forget particles. Think fields, like Maxwell’s electromagnetic field. A field can always be decomposed into an infinite sum of elementary sine waves (this is the essence of a Fourier-transformation). We know how to apply the rules of quantum mechanics to these sine waves: They are so-called harmonic oscillators. And a quantum harmonic oscillator has very interesting behavior: its energy levels are quantized. Its energy increases and decreases one quantized unit at a time.

So now that we decomposed the field into quantum harmonic oscillators and recognized that the energy levels of these oscillators are indeed quantized, comes the next step: We associate these energy levels with what we perceive as particles. So let’s introduce another field, the “electron field”. It interacts with the electromagnetic field. Both fields are quantized. As a result of these interactions, excitations of the electromagnetic field may turn into excitations of the electron field or vice versa. As it turns out, this picture correctly describes quantitatively (!) phenomena such as the emission or absorption of a photon, or the creation or annihilation of electron-positron pairs.

This all happens between observations. So no wavefunction collapse is involved. The fields that we are talking about are not the wavefunction. They are operator-valued quantum fields, those (mathematical) operators acting upon the wavefunction. Ultimately, the way they act upon the wavefunction can be used to determine the likelihood of detecting a particle at a certain place with specific properties. That part works the same way as in ordinary quantum mechanics. It’s how we get to that point, using a theory that can describe particle interactions (and which, incidentally, is also fully relativistic and causal) that is different.

Are wormholes a fact, or just a belief? Is there any theory that indicates that wormholes are possible to exist, and can be opened artificially? Can wormholes occur naturally, or the only way for them to exist, is to be created artificially?

OK, allow me to clarify a few things.

A fact in physics would be something that we observe. We have not observed wormholes.

A theory (i.e., a body of established knowledge) can predict things that have not, or have not yet, been observed. Wormholes are a prediction of general relativity. Or to be more precise, wormholes represent a class of possible solutions of Einstein’s field equations of gravitation.

However, “possible” in this case has to be seen with a caveat. The solutions are possible in the sense that they are mathematically valid solutions of the equations. But are they possible physically? That is, are they consistent with the nature of things that are not part of general relativity theory, such as the known nature of matter?

The answer is, probably not. Wormholes, especially stable (traversable) wormholes may only exist in the presence of matter with negative mass. While such matter is allowed by the rules of general relativity, in quantum field theory, negative (aka. “exotic”) matter means an unstable vacuum, and that would be very bad news for our universe. Wormholes also may violate causality (e.g., by allowing time travel) that also contradicts our understanding of the causal nature of this universe.

These are all reasons to believe that wormholes do not exist, but allow me, then, to make a point about the word “believe”. In this context, it does not mean something like religious belief. It is not unconditional acceptance of an article of faith in the absence of hard evidence. Rather, it is an (admittedly sloppy) expression stating an informed opinion, a likelihood if you wish, something that is not proven with mathematical rigor or not observed, but what is likely to be the case based on the expert’s prior experience with theory and observation. That doesn’t mean that the expert cannot be wrong; quite the contrary, we call it a “belief” precisely because it is subject to change in the light of new evidence.

Finally… even if we find, despite all of the above, that wormholes exist in this universe does not imply that they can be created. That may not at all be possible; or, it may require steps, such as manipulating matter or the scale of entire stars or larger, that will likely remain forever impractical.



{{How close are physicists to solving the mystery of dark matter and dark energy?}}
We won’t know until we know. Seriously.

There have been numerous experiments trying to detect dark matter directly. To date, all these experiments produced either negative results (only upper limits on the rate at which dark matter may interact with normal matter, but no actual detection) or detections that were not confirmed by other experiments and didn’t reach the statistical certainty needed to claim a discovery. (In other words, the data may just be random noise.)

On the other hand, modified gravity theories, which attempt to do away with dark matter, have also failed to deliver a result that would be widely accepted. In my personal opinion* the field is hindered by the famous “MOND” (MOdified Newtonian Dynamics) that is an ad hoc formula designed to replicate the anomalous rotation curves of galaxies but accomplishes little else, violates even basic conservation laws, still requires some form of dark matter on the cosmological scale, and nowadays, it is increasingly presented with equally ad hoc modifications such as the famous “external field effect”. Unfortunately, those who do not work in the field find it hard to distinguish MOND from properly constructed, relativistic field theories of gravitation, and often use MOND as a straw man to criticize modified gravity theories in general, not realizing that many of the shortcomings they address are specific to MOND. But even if I put MOND aside, no modified gravity theory can claim full success on all fronts; even the more successful theories struggle when confronted with the totality of observational data on all scales (ranging from the lab, through precision solar system tests, all the way to astronomical and cosmological observations).

All of the above is about dark matter; dark energy is an even tougher case. The fundamental problem is that there are several “easy” candidates for dark energy, but no known way to distinguish them experimentally. If there is a cosmological constant, for instance, then it’s just that, a constant of nature. Or could it really be the zero-point energy of quantum fields, but with a much lower energy cutoff than the Planck scale value, which is itself suggested by the somewhat ad hoc assumption that quantum field theory is “just an effective theory”, so the summation must stop at the scale where the theory is believed to lose its validity? Or is there perhaps a yet to be discovered so-called scalar field, which yields a self-interaction potential with the right dark energy like behavior? We do not know.

So really, we don’t know how close we are, because we don’t really even know what we don’t know. We still have much to learn.

*Having worked on modified gravity myself, I am knowledgeable on the topic but not necessarily unbiased. Hence this disclaimer.

Could a human enter a black hole to study it?

In principle, as in a thought experiment, the answer is yes, but with an all-important caveat: there is no way out. Literally no way out. It’s not that there is a way and it takes a clever human to find out. It’s not that it is difficult. No, getting out of the black hole would require (literally) the ability to travel backwards in time. So the human who was foolish enough to enter the black hole is condemned.

There is another caveat: All this is about a classical black hole, ignoring quantum field theory, Hawking radiation, and speculative considerations about a “firewall”. These issues may prevent a human from entering a black hole because either the event horizon just isn’t there or the human is destroyed before reaching it.

And speaking of being destroyed… that’s just one of the many practical obstacles. For starters, the nearest known black holes are thousands of light years from the Earth. The farthest a human being has been from the Earth to date is roughly one light second of distance (the Moon). The farthest any manmade object traveled to date would still be under a light day (Voyager 1) and it took that spacecraft more than 40 years to get there. So given our technology at present, it would take millions of years to reach the nearest known black hole.

Furthermore: A black hole with a mass no more than a few times the mass of the Sun is very compact, at most a few dozen kilometers wide. When you get that close to a source of strong gravity, tidal effects become gigantic. These will rip any spacecraft, any human to shreds (this is what’s called whimsically “spaghettification”) long before they even get close to the event horizon.

The only black holes that have gravitational fields that are gentle enough near the event horizon are the largest supermassive black holes (not puny ones like “our” supermassive black hole in the central region of the Milky Way but much larger ones in other galaxies.) The nearest such very large supermassive black hole is tens of millions of light years from here. To get there with the technology we have… well, we can’t. We have no technology yet that would allow us to escape the gravitational pull of our own galaxy, even if I ignore the enormous (billions of years) travel times involved as a mere “practicality”.

Why is gravitational constant G important in dark matter calculation while the general theory of relativity shows that there is no force but curvature of space-time?

Einstein himself described the geometric interpretation of gravity as a useful mental aid, nothing more, so I would certainly avoid characterizing gravity as “no force but curvature”. It certainly feels like a force to me when I hold a brick in my hands; even more so when I accidentally drop that brick on my big toe. Even if we accept that this force is present only because my reference frame here on the surface of the Earth is noninertial, it is still a force, or at least, a pseudoforce, like the centrifugal force.

Putting that aside, the gravitational constant is a coupling constant. It describes the coupling between matter fields and the gravitational field. Whether I interpret that gravitational field as geometry (i.e., the metric of spacetime) or consider it a physical field (which it is, as it carries, after all, energy and momentum just like the electromagnetic field) is irrelevant here. What matters (and the reason why we feel gravity) is that there exists a coupling term between this field and other fields in the Standard Model of particle physics, and the strength of this coupling is determined by the gravitational constant.

Dark matter is expected to follow the same rules as normal matter. (If it didn’t, that would mean the end of interpreting gravitation as the one-and-only geometry of spacetime.) So its coupling to the metrical field of gravitation is governed by the same coupling constant.

Indeed, all that energy was available in the early universe.

When light atoms fuse, the potential energy between their constituent particles (which is negative) is offset by a release of (positive) kinetic energy. This kinetic energy usually appears as the kinetic energy of photons (electromagnetic radiation) that is released by the reaction.

When accretion disks release energy, once again it is the negative gravitational potential energy of infalling matter that is offset by the positive kinetic energy that is gained by the infalling particles; these particles then collide with each other, the kinetic energy is randomized as heat, and it is released again as electromagnetic radiation. Additionally, some of the kinetic energy is transferred to a magnetic field as the cloud of charged particles rotates and that, too, ends up eventually as the kinetic energy of particles that this magnetic field accelerates.

The common theme in both these cases is that the total energy need not even be anything other than zero! Ignoring their rest masses (which do not figure in the reaction anyway) when a bunch of protons are far away from each other, the potential energy between them is zero. As they are brought together, the potential energy at first becomes positive (due to electrostatic repulsion) which means that energy must be invested to push them together; but once an energy barrier is reached, the potential energy rapidly drops and becomes negative, the protons fuse, and the excess energy is liberated as radiation. The sum remains zero: the positive kinetic energy of that radiation is balanced exactly by the negative potential energy of the bound state.

And this leads me to a concept favored by some physicists, which is called whimsically “the ultimate free lunch”: the possibility that the total energy content of the universe is exactly zero and has always been.

This is not necessarily a silly idea but may not be very meaningful either. While we can deal with quantities like energy densities or the energies of finite systems, the total energy of the universe may be a mathematically ill-defined notion. Still, I think it is an intriguing idea that perhaps throughout it all, the energy-content of the “primordial soup” and the (negative) gravitational binding energy of the universe were always in perfect balance, and thus the universe came free of charge, “the ultimate free lunch”.

No cavities are involved. Do not confuse the theoretical model that is sometimes used to illustrate blackbody radiation with the actual, physical process that produced the CMB.

The CMB is, in fact, thermal radiation that was given off by hot, incandescent gas. When the universe was approximately 385,000 years old, give or take, the (mostly hydrogen and helium) gas that was its matter content cooled down to a temperature where it was possible for its atomic nuclei and electrons to recombine into electrically neutral atoms.

The result is that the previously opaque, glowing plasma gradually became transparent. Its own residual incandescent glow, blackbody radiation at a temperature of approximately 3000 K, was now able to propagate unimpeded over greater and greater distances. When the process was complete, the result was a thin, neutral, transparent gas and this residual radiation, light originating from every point in space and traveling in every direction. As the gas was now transparent, it no longer interacted with this light; the gas and the light “decoupled”.

Fast forward 13.8 billion years to the present. The gas eventually coalesced into galaxies, stars, planets, underwent nuclear fusion and cataclysms like supernova explosions that produced heavier elements, eventually resulting in rocky planets, even life. The residual light is still there, redshifted as a combination of expansion and changing gravitation, by a factor of about 1,100; its blackbody temperature now corresponds to 2.7 K instead of 3000 K.

But it is still the same incandescent glow that was given off by a cooling mix of ionized hydrogen and helium as it recombined with electrons to form a neutral gas.


I think that even with all the cosmological knowledge that we possess, speculating about the extreme far future is a bit… premature. Unwarranted.

But first… stars in the big scheme of things are not that important, really. Why would the cosmos as a whole care that some blobs of gas condense under their self-gravity and for a while are hot and dense enough for nuclear fusion? There are so many, so many other things that can take place in this universe within the confines of the laws of physics that we know. Just because stars are important to us, their existence is not really all that relevant on the cosmic scale of things.

Anyhow… the universe is presently just under 14 billions of years old to the best of our knowledge. It really is a young universe. There are dwarf stars that have enough fuel to last a trillion years or more. And even that far into the future, there will still be enough hydrogen for new stars to form, albeit at a much reduced rate compared to today.

But let’s be bold and venture into the really far future. A quadrillion years? More? The actual numbers become kind of meaningless at the point. There will indeed be a time in the extreme far future when there will no longer be stars, only cold stellar remnants. This future universe is far from being a void yet. It will still occasionally see, e.g., dead stars collide, perhaps even produce supernova like explosions momentarily shining with the brightness of an entire (present-day) galaxy.

Going even further into the future we will see a universe, still expanding, with distances between (dead) galaxies now so large, even if they were full of luminous stars, they would still be invisible to each other. And these galaxies would slowly “evaporate”, ejecting (dead) stars as a result of three-body gravitational interactions, losing them altogether. Their supermassive black holes would continue to grow… for now. But after an insane number of years (now we’re talking about numbers measured using up to a hundred digits or more) smaller black holes begin to vanish due to Hawking evaporation. And then larger black holes would follow suit, no longer fed mass-energy from the vanishing cosmic microwave background.

Ultimately, if our understanding of gravity and quantum field theory is correct, even lumps of matter that are too small for classical gravitational collapse would quantum tunnel into black hole states and, through that, into Hawking radiation. That radiation will then be redshifted into nothingness by the expanding universe. Ultimately, the void remains.

Or not.

For starters, the timescales are now insane. Thousands of digits measuring the number of years. Or rather, thousands of digits measuring the number of digits measuring the number of years. Numbers that are impossible to conceptualize. (At this point, it really becomes irrelevant if we use gigayears or nanoseconds to measure time… the number of digits of the number of digits would barely register a change.) Over such immense timescales, even very rare quantum events are possible. Such as the spontaneous emergence of a whole new universe through a Big Bang like event.

Or maybe none of this happens. Maybe concerns that the Higgs vacuum in a renormalizable quantum field theory is metastable are valid, and that after a long but much more finite number of years, the whole universe “snaps”, undergoing a neverending phase transition as the vacuum decays into a lower energy state, only to find that this state is not bounded from below, and thus the decay becomes perpetual, a stable vacuum no longer possible.

Or, far more likely, everything I wrote above is a fancy tale of fiction, a result of an unwarranted extrapolation of what we know today over countless orders of magnitude.

What we do know is that we are probably safe for the next billion years or so. That’s already about 100,000 times longer than all the known history of human civilization. A billion years from now, if we are still around, we’ll have to find a new home as the Earth becomes uninhabitable. But the universe as a whole? It will still look pretty much exactly the same as it does today, as a billion years is just a blink of an eye in the big scheme of things.

Originally Answered: Could our 4D space be thought of as a curved surface embedded in a flat 5D hyperplane? Would math predict physics as we currently understand it?

Oh yes, of course we can think of a 4D manifold as being embedded in a higher-dimensional manifold.

But the thing is… we don’t have to.

The reason is that there is a distinction between intrinsic and extrinsic curvature, and in gravitational physics, we only care about the former, not the latter.

To illustrate the difference, think why you can roll a sheet of paper into a cylinder shape but not into a spherical shape. In both cases, you introduce curvature. But you can form the cylinder without stretching or compressing that sheet of paper. Distances between points on that sheet remain the same.

In contrast, you can only turn something flexible, like a rubber sheet, into a spherical surface. You need to stretch that rubber sheet. Distances change as a result.

Our best gravitational theory to date, Einstein’s general relativity, associates the gravitational field with the intrinsic curvature of spacetime. This curvature exists independent of any higher dimensional space into which our 4D manifold might or might not be embedded; it is measured by how distances change in spacetime (i.e., the metric of spacetime.)

In short, whether or not our spacetime is embedded in a higher dimensional space and how it appears there (i.e., its extrinsic curvature) seems to have no bearing on observations in physics. As such, from a theoretical perspective it looks like an unnecessary, superfluous assumption.

Black holes have mass but not matter. Can someone explain why and how does it work?

Let us distinguish two completely different things:

  1. Astrophysical black holes
  2. Theoretical vacuum solutions of Einstein’s general theory of relativity.

Regarding #2, the simplest solution of general relativity, the Schwarzschild solution from 1916, is a vacuum solution that is static (not changing with time) and spherically symmetric.

When it comes to #1, the following points can be shown using mathematics: a) the so-called Oppenheimer—Snyder solution from 1939, demonstrates how a spherical cloud of pressureless matter (“dust”) collapses in such a manner that ultimately, the limiting case (in the far future) is the Schwarzschild solution; and b) even if the original cloud deviates from spherical symmetry, the result is still the Schwarzschild solution if there is no net rotation (or the Kerr solution, which is an axisymmetric, static vacuum solution, if the cloud has net rotation.)

Finally, there is another important theorem in general relativity, Birkhoff’s theorem, which tells us, among other things, that outside a spherically symmetric mass (or far enough from a mass so that its deviations from spherical symmetry don’t matter) spacetime is the same as in the Schwarzschild solution.

Putting these pieces of knowledge together, we can deduce that when a cloud of matter collapses under its self-gravity, it can be treated more or less as dust (since gravity proved stronger than pressure, so as the collapse progresses and gravity becomes even stronger, pressure becomes irrelevant); and that although the “end stage” will never be observed by outside observers, as a result of collapse and extreme gravitational time dilation and redshift, the collapsing matter rapidly vanishes from sight, leaving behind an apparent “hole” that, for all practical intents and purposes, is indistinguishable from a Schwarzschild (or Kerr) black hole.

So real black holes have mass and matter. The idealized end stage is a vacuum solution, but this is arguably mathematics, not physics (as it describes “physics” in the infinite future as seen by outside observers.)

Today I saw a trivia question, which asked what the shape of the universe was and answered "it resembles a vuvuzela". I assume they misinterpreted something, but how did they come to this belief?

The universe does not have a shape.

In the so-called standard or concordance model cosmology (the Lambda-CDM model), space is “flat” on average, infinite, and characterized by Euclidean geometry; however, the average density of matter in this infinite space is changing over time as matter flies apart.

The rate of this expansion was rapid at the beginning (perhaps characterized by a very brief period of extremely rapid expansion — cosmic inflation) then it was slowing down for about 8–9 billion years; in the past 5 billion years or so, it has been accelerating again.

When plotted on a sheet of paper as a function of time, the curve that characterizes the rate of expansion can indeed vaguely resemble the shape of a horn or vuvuzela.

But that is not the “shape of the universe”. Unfortunately, this vuvuzela shape is often used to depict the universe in popular accounts (magazines, documentaries), sometimes using very convincing, elaborate 3D graphics, which might explain the origins of the trivia question.


PS: Questions like this, of which there are many here on Quora, remind me repeatedly how innocent attempts to offer intuition into some of the deep mysteries of physics — e.g., the “rubber sheet” analogy used to illustrate gravity, the “vuvuzela” shape to show the expansion of the universe, the “miniature solar system” shape of an atom — can badly backfire, and instead of offering useful clues, often accomplish the opposite and become obstacles to understanding. I think this is an important lesson for anyone who tries to write about science to a broad audience.

Moments in time don’t “usually happen to everything”. This idea, absolute or universal time, was demolished by relativity theory. Today, it is well-established experimental fact that time is always observer-dependent.

Let me give you an example. Today, we have atomic clocks that can measure time with such exquisite precision, if such a clock had been ticking since the beginning of the Universe, 13.8 billion years ago, it would still not have gained or lost a full second.

Now you might think, okay, if we have such precise clocks, we should keep then in sync. But we cannot. The clock at the top of a building will tick slightly more rapidly than the clock at the ground floor. A clock over a cave in the ground will tick slightly more rapidly than a clock over a large deposit of some heavy ore. In fact, these clocks are so sensitive, they even have potential military applications: as the rate at which the clock ticks changes when we move the clock around, it can be used to map, e.g., underground bunkers.

The differences between clocks is tiny here on the Earth, because the gravitational field of the Earth is relatively weak and our clocks don’t move about very rapidly either. All this changes near a black hole, where gravitation reaches its theoretical extremes. A clock near the event horizon ticks very, very slowly compared to a clock far away. At the event horizon, the clock would actually come to a halt as seen by a distant observer.

Think about what that means. It basically tells you that if you watch a clock (or any other object) approach the event horizon, it will appear to move, and tick, ever more slowly. The moment when it reaches the event horizon? That will take forever for you to see. In fact, you never get to see the event horizon itself form: it, too, remains forever a future moment in time for you.

Of course for a clock, or an observer, who actually falls into the black hole, that is not the case: it will reach the event horizon in a finite amount of time. But even for this observer, the event horizon is a moment in time: once crossed, it is a moment in time in the observer’s past. (Which explains why that observer cannot ever get out again: it would require traveling backward in time.) And the singularity? That is an unavoidable future moment in time for this infalling observer. For the observer outside the horizon? Recall that the horizon itself never forms: to that observer, the singularity doesn’t exist, as it remains forever hidden in the future, behind that event horizon that he will never get to see.

All this is very counterintuitive, and all this happens in a spherical region of space that is characterized by the gravity of the black hole, hence our tendency to think of the event horizon as a sphere and the singularity as a location inside that sphere. This is why it is important to keep in mind that we are talking about extremes of general relativity here: space and time are malleable, our intuition of absolute time and Euclidean space are useless in this context.

First, the total mass of the universe is not known. True, in the standard cosmological model, we have a universe that is spatially infinite and homogeneous (same everywhere) so indeed, its mass would be infinite, but this is by no means a certainty.

Second, even in this infinite universe, the mass density (mass per unit volume) remains finite.

Third, there is the shell theorem, and its relativistic extension, Birkhoff’s theorem. The gravitational field inside a uniform spherical shell of matter is zero. If the Earth were hollow, you could float weightlessly inside. So even in an infinite universe, concentric shells of matter surrounding you will not have a gravitational influence on you. Only local deviations from this homogeneous background (e.g., the presence of a nearby star or planet) matter.

And fourth, there is the issue of time. The universe has a finite age. This defines the range within which matter could have influenced us. Matter outside this “cosmological horizon” just hasn’t been around long enough to exert an influence on us (or us, on that lump of matter.)


Modifié le: jeudi 4 février 2021, 17:34