In physics and engineering, the term “continuum” implies that matter (solids, liquids, and gases) is treated as a continuous distribution of mass, even though at the microscopic level it is made up of atoms and molecules. The idea is to ignore the atomic structure and model materials as if they can be divided endlessly. For example, when analyzing stress in a steel beam, you don’t model every atom—you assume it’s a smooth material with properties like density and elasticity defined at every point. Two most popular examples of the concept of are:
- the fluid dynamic continuum for modelling liquids and gases and
- the space-time continuum used in General Relativity.
In #1 the fluid is modelled using the Navier-Stokes equations which are Newton’s Laws of Motion applied to small chunks of the continuous medium. In #2 space and time are modelled using gravitational equations. whereby massive objects distort space time and their motions, in turn, are determined by distorted space-time.
A continuum alone can never exhibit interesting behaviour!
All the interesting behaviour in a fluid continuum, such as jets, vortex streets, breaking waves, and so on, only occur near the boundaries of the continuum, i.e. where the fluid flows past a solid object or where one fluid meets another at a surface. Left to itself a fluid can only progress to a maximum entropy state, a sort of Heat Death of the Universe.
This is intuitively obvious for a fluid continuum but it must also apply to other continua such as space-time. General Relativity is pure geometry and has no mechanisms to give rise to interesting behaviour. Consequently, although General Relativity predicts the existence of black holes , it has little to say about their behaviour.
In a manner reminiscent of Gödel’s Incompleteness Theorem, General Relativity predicts its own boundaries. The event horizons of black holes are predicted but they are not continuous and differentiable. So, by definition, they cannot belong within the space-time continuum itself.
General Relativity predicts its own incompleteness.
I’ve been thinking of moving black holes, switching from Bronte Park to New Norfolk.
I discovered a great coffee/bookshop full of literary types and not a physicist among them.
If you mean Al;exander’s Black Swan, the occasional physicist has been seen there. Sorry.
Hi John,
Black holes have been more topical than usual lately, not sure why, maybe some theoretical discoveries in the science news, so no surprise you have been thinking about them.
My knowledge of this is limited, so unlikely my comments add anything of value. I am not conversant with Navier-Stokes or Godel. My mind is stuck there with Newton.
But your comment that certain observations and surmises are “intuitively obvious” aroused me.
The reason such phenomena seem intuitive reminds me of Feynman’s comments, implying that nothing is intuitive about the quantum world. He said something like if you think you understand it then you don’t. Now your essay is about continua, the opposite of quanta, so that may be why such things are intuitive.
Is it true that if it were indeed possible and practicable to model matter at the quantum level, the results for macro matter such as fluids in a river or airstream or weather pattern, would be the same? Would tracking the motion of every atom, integrated generate the same behaviour of the matter at the macro level? Intuitively it would!
So your statement that the event horizon of black holes are not continuous and differentiable should be true using methods (equations, theorems, intuitions) that are from the maths of continua. But at the event horizon, is not what happens next predicated on events at the quantum level? So if the analysis reverts to quantum methods, does GR still predict its own incompleteness?
In other words it is dangerous to expect phenomena to be intuitive at its real nature which is not continuous.
Just saying.
I’m confused, “Nothing can travel faster than the speed of light”, so two photons on a collision course can not hit at a speed more than the speed of light?
Both photons are travelling towards each other at the speed of light. So they approach each other at twice the speed of light. But neither one of the photons, not anything else, is actually travelling faster. The distance between them is closing at a greater than light speed, but that is not a thing travelling.
Now when my friend Sabine travels past a shoe store, she reverses back at a speed that has been measured at several orders of magnitude faster than light.
Arthur,
Two photons can’t collide so they can’t be on a “collision course”. They are just waves which just go past one another. They only interact with atoms and electrons so there is no experiment which could answer this apparent anomaly.
Photons are mass-less particles which behave as waves as and when they feel like it. They are not affected by gravity, but move, at the speed of light, through time-space manifestation which itself is influenced by gravity. Gravity’s influence upon a body within the space-time continuum and upon this continuum itself dimishes/decays exponentially with greater distance from the body in question or the location in space-time observed. Space-time is affected by gravity because it has mass/energy.
When photons meet, they merge for a Planck/quantum second and go on their way, having exchanged the gossip they picked up since the Big Bang. This gossip (information) we can only obtain as (en)light(ened) beings. Us ordinary folk have no hope of seeing the light for what it really is.
Allan
Different physical laws predominate at different spatial and temporal scales. Quantum effects predominate at subatomic scales. We are talking about the spatial scale of stars not atoms. Feynman’s comment concerns only small scales. The fact remains that the polar jets of black holes are the most continuously energetic objects we have yet observed and, thanks to the dead end created by General Relativity, there is still no accepted explanation for this. An unconvincing attempt was made by Hawking who postulated new particles and a new sort of radiation, neither of which have been observed but who cares? Once you get into the mystical claptrap of quantum mechanics anything goes.
My view is that the polar jets are excited by the gravitomagnetic shock wave generated when a star falls below the event horizon. See Black Hole Accretion Shocks on Blackjay.
Your comment on modelling matter at an atomic level and seeing what macroscopic results emerge is spot on. They did this with radiation a century ago by treating atoms as emitters and absorbers of radiation then adding up the radiation from a hot body at a constant temperature (a “black body”). Far more radiation in the ultraviolet was predicted than was actually observed – “the Ultraviolet Catastrophe”. Planck showed that it was caused by the fact that radiation is only emitted and absorbed in discrete little packets called “quanta” where the energy of a quantum depended on frequency not amplitude.
Yes, and each quanta are only absorbed by the matter it is interacting with if its frequency matches the energy level for an electron (to jump up to the next level). If there is no match, then that quantum at that frequency is not absorbed, so the photon continues through the matter. In glass, visible light frequencies match almost none of the electron energy levels, so continue through, making glass transparent. In another case, if say only the red frequencies are absorbed by particular atoms in a sample of matter, then the colour to our eyes is the rest of the spectrum, not red.