First read this piece by St. Feser:
So you think you understand the Cosmological Argument?
about the traditional structure of Cosmological Arguments, rebutting several popular misconceptions.
Now take the following argument scheme, making suitable choices as needed:
- (Major Premise) Every [thing/event] with property X needs a [cause/explanation/reason] outside of itself to [cause/explain/be the reason of it]
. - (Minor Premise) There is at least one [thing/event] with property X.
. - (Inductive Principle) You have a choice...
A. argue that an infinite regress of [causes/explanations/reasons] for the X's is unreasonable, OR
B. argue that such an infinite causal chain would itself have property X, OR
C. argue that the entire set of X's taken together (which might, depending on X, include the entire physical universe we know and love) has property X.
. - (Conclusion) Tracing back the [causes/explanations/reasons] back to their ultimate origin, we find that there is [one/at least one] thing which does not have property X, which, taken [singly/together], [causes/explains/gives the reason for] all the things which do have property X.
. - (Atheist Baiting) Add the famous words: "And this all men call God". Works best if ~X is a traditional divine attribute, or even better if you can collect several such ~X's and can argue that they all refer to one and the same Exalted Being!
For example, in the debate, St. Craig's kalam argument used "comes into existence" as X, and then used a lumping strategy (3C) to talk about the universe as a whole and ask whether it had a cause. This form of the Cosmological Argument ended up being strongly dependent on what the Science of the Big Bang actually shows, but most forms don't really depend that strongly on Science.
Other traditional X's include "changing with time", "contingent" (something that might or might not exist), "composite", and some other possibilities mentioned in the link above. The idea is that there are some features of objects which make us seek out causes for them, for example if an object is composed of several disparate objects, we naturally want to know what brought them together. Depending on what you pick for X, the Cosmological Argument may be more or less plausible.
You will also want to consider what type of causal concept you want to include in your argument. A key question is how we know there is such a thing as causation? If it is primarily for empirical reasons, then presumably we know about it through some type of inductive argument from experience, in which case we could wonder how applicable it will be in unusual situations. On the other hand, if it is primarily motivated by reason, through analyzing what types of explanations would make sense of the universe, it may be less dependent on observation. Or perhaps the truth is somewhere in the middle.
Another thing to figure out is what types of entities are connected by cause-effect relationships. Does a cause have to determine the effect with certainty, or is it sufficient if it in some way produces it? For example, if we want to argue that all contingent things were caused by something which is necessary, this is a contradiction in terms unless a necessary thing can produce contingent things, i.e. if causes don't have to be deterministic. A related question: when we talk about causes, are we primarily talking about beings causing things to happen (a.k.a. agent-causation), or states of affairs causing things to happen (a.k.a. event-causation), or both?
Regarding step (5), note that excessively glib atheist baiting obscures the fact that nearly everyone should accept some type of Cosmological Argument, even if they don't necessarily take it to a Theistic conclusion! If you are going to talk about causes/explanations/reasons AT ALL (and I really don't see how to avoid this) then you really need an account concerning the domain to which the concept is applicable. And then it is an interesting fact, that either you must accept infinite or circular chains of [causation/explanation/reasons] or you end up going outside the domain to something else which is different.
This type of reasoning should be interesting, even if you are an atheist. The trouble is, if people only encounter Cosmological Arguments in the context of Theism, then Atheists adopt an argumentative approach where they just feel the need to poke a few holes in the arguments and then retreat to where they were before. This doesn't do justice to the fact that there are numerous X's for which the argument's premises are at least plausible, even for people who don't start out committed to any particular religious doctrine.
For example, Carroll himself gives an account of the scope of causation when he says:
Why should we expect that there are causes or explanations or a reason why in the universe in which we live? It’s because the physical world inside of which we’re embedded has two important features. There are unbreakable patterns, laws of physics—things don’t just happen, they obey the laws—and there is an arrow of time stretching from the past to the future. The entropy was lower in the past and increases towards the future. Therefore, when you find some event or state of affairs B today, we can very often trace it back in time to one or a couple of possible predecessor events that we therefore call the cause of that, which leads to B according to the laws of physics. But crucially, both of these features of the universe that allow us to speak the language of causes and effects are completely absent when we talk about the universe as a whole. We don’t think that our universe is part of a bigger ensemble that obeys laws. Even if it’s part of the multiverse, the multiverse is not part of a bigger ensemble that obeys laws. Therefore, nothing gives us the right to demand some kind of external cause.
There seem to be some question-begging moves in this paragraph, but leave that aside. My point is that Carroll gives a positive account of when he thinks makes the notion of [cause/explanations] make sense. He endorses a version of (1) whereby the concept of causation makes sense if (a) there are laws of nature, understood as unbreakable regularities, and (b) there is a thermodynamical arrow of time whereby entropy increases, making a distinction between the past and the future. (Since causes normally precede effects, but the laws of physics don't strongly distinguish between the two directions of time except through thermodynamics, it seems clear that the arrow of time has to play some role in distinguishing causes from effects in physics.)
He also allows (2) that this concept—though not fundamental in his opinion—nevertheless makes sense for certain particular cases.
Then for (3) he allows us to lump together the universe taken as a whole, but claims that this whole does not meet his criterion (1). He thus comes to an object—the whole universe, apparently—for which, in his view, it wouldn't make sense for it to have a cause (4), although he does not identify it with God (5). Thus his reasoning has an implicit atheist version of the Cosmological Argument behind it. Though one can certainly question whether the metaphysical assumptions behind this claim are right.
But it's just possible you came here hoping, not to construct your own Cosmological Argument, nor to deconstruct Carroll's, but instead to find out what I think about it, something which you may think I have postponed saying for quite long enough. Well, it just so happens that I've written a 16,000 word essay on the Cosmological Argument and related topics, and will be posting it in installments over the course of the next few weeks.
In line with the whole "conrstruct you own cosmological argument" idea is this website from Alexander Pruss and Joshua Rasmussen: http://necessarybeing.net/
Hhhmmm I wonder if the universe could be eternal in a timeless way, prior to the Big Bang?
Charlie,
What do you mean by "eternal in a timeless way"? And in what sense would it then be "prior to the Big Bang"? I see what these would mean in the case of God, but I suppose you're thinking that this would be some type of physical model?
Dr. Wall, how do you believe Dr. Carroll would justify his first premise {whereby the concept of causation makes sense if (a) there are laws of nature, understood as unbreakable regularities, and (b) there is a thermodynamical arrow of time whereby entropy increases, making a distinction between the past and the future. (Since causes normally precede effects, but the laws of physics don't strongly distinguish between the two directions of time except through thermodynamics, it seems clear that the arrow of time has to play some role in distinguishing causes from effects in physics.)}? It seems to me you're right about the question begging against other types of causation besides his own which i'll call physical event causation.
I thought maybe if time was relational between things, the universe-as-a-whole would be timeless or maybe if time wasn't fundamental (like holographic) maybe there would be some timeless somewhere.
I don;t know how it can be "prior" i guess it can't.
But thanks Aron, I've learned a lot from you.
Charlie,
Thanks, that tells me a bit more about what you were thinking.
I'm not sure I understand your first idea; there is a sense in which one can always describe the universe in a timeless way by modelling time as a coordinate; or rather, I guess I would say that there is a sense in which everything exists timelessly (that is, they exist) and when we say that something exists at a particular time, we are merely specifying the coordinate vicinity in which it exists. A-theorists like St. WLC would think this leaves something important out, but I disagree...
Regarding your second idea, it could well be that there is some type of "dual description" of the world where it is described in terms that don't directly refer to time. In that case we might say that time is "emergent", especially if the dual description were more complete, and the temporal description is just an approximation valid in certain contexts. (If both descriptions were equally valid and isomorphic, one might instead regard them as equally fundamental, or at least one would need a good argument to privilege one of the two descriptions over the other.)
I originally thought maybe you had a different idea, where the universe has two parts, one of which is timeless and the other temporal, such that the first part affects the second part but not vice versa (hence it is "prior"). I guess one could argue that the laws of physics are like that...
I'm also reminded of Sydney Coleman's paper on the effects of baby universes (behind a paywall, but you should be able to access it in a major academic library):
Black holes as red herrings: Topological fluctuations and the loss of quantum coherence
which includes the rather remarkable statement that:
John,
The idea that causation is just a name for constantly conjunction or regularities was introduced into Philosophy by David Hume, and you can read his original arguments for it in his book An Enquiry Concerning Human Understanding. Since then there have been numerous discussions pro and con, which you can find reviewed in various places on the internet.
Carroll's views are also likely influenced (perhaps indirectly) by the writings of Bertrand Russell, who argued that the concept of causation is not ultimately meaningful. Of course all of this is extremely controversial.
Regarding the arrow of time and its relation to causality, I imagine Carroll goes into more detail about the reasons for his views in the book he mentioned in the debate (I think he means this one) although I haven't read it so I don't know how much he delves into the philsophical issues. Of course, he's also written numerous articles on the arXiv on the arrow of time. When I wrote that
I was trying to acknowledge that at least this part of what he's saying is grounded in fairly established physics. The universe "as we know it" started in a much lower entropy state, and this is the reason for the Second Law, which in turn explains why "causes" normally preceed "effects" in our zone of the universe. What we don't know is why the universe started in a low entropy state. And Carroll does not regard this as a meaningless question, rather he has devoted a fair amount of his research to trying to construct models in which this fact is (or seems to him to be) more natural.
Professor Wall,
Great blog post. Here’s what I have to contribute:
P1: Every being or fact (thing or event] that is contingent (Property X = contingent) must have a cause or explanation.
P2: There is at least one being or fact that is contingent.
P3: Such cause or explanation cannot run into an infinite regress.
C1: It follows there must be a First Cause.
C2: Obviously, the First Cause must itself be uncaused.
C3: This means there is at least one being or fact that does not have Property X and hence, must have the Property of necessity rather than contingency because of P1.
This necessary being is what is meant by God. (In the Abrahamic faiths, there is only one God.)
( Ref: Modern Physics and Ancient Faith, Stephen M. Barr)
I did not find P2 (the minor premise useful in this argument for God's existence.
Would like to have your comments and from all.)
TY
Hi Dr Aron,
An alternative formulation belowt:
1. Every being or fact (thing or event] that is contingent (Property X = contingent) must have a cause or explanation.
2. There is at least one being or fact that is contingent.
3. The Universe is a contingent fact.
4. Therefore the Universe has a cause/ explanation for its existence.
5. For a thing with Property X, such cause or explanation cannot run into an infinite regress (no turtles all the way down.).
6. It follows there must be a First Cause.
7. This means there is at least one being or fact that does not have Property X but instead, the Property of necessity.
8. This necessary being is what is meant by God (at least in the Abrahamic Faiths)
9. The explanation of the contingent universe must be this necessary God.
TY.
TY,
The main challenge is to explain how you know that P1-P3 are true. How would you justify them to somebody who rejects one or more of them?
You say you didn't need the minor premise in your first argument, but if you don't have it, how do you prove that the necessary thing is a "First Cause"? Maybe everything is necessary, and nothing causes anything else. Sure, if you just want to prove the existence of something that is necessary, you only need the weaker assumption "P2: At least one thing exists", a premise which even the most stubborn atheists are generally willing to accept. But if you want to prove the existence of a necessary "First Cause" which is not the same as what we see, but explains it, then I think you need some premise like this. In your second argument, premise 2 is indeed redundant in the sense that it follows from premise 3, although any argument that premise 3 is correct might well need to start with premise 2 as a stepping stone.
By the way, you should also ask if a necessary being can cause another necessary being to exist. In his Third Way, St. Thomas actually has to run through the argument twice, since he thinks some necessary beings might have their necessity caused by another. So then he traces these back to a being which exists of its own necessity.
Also, if we are willing to assume the Abrahamic faiths, then we already know there is a God who created everything else without needing the Cosmological Argument. So you could also ask how you would argue that the First Cause is likely to have other divine properties associated with God.
Prof. Wall,
Thank you for the comments on the formulations of the Cosmological Argument or Argument from Contingency. Allow me take another stab at the Cosmological Argument and the most troublesome premises:
P1: The universe is contingent (meaning, caused to exist by “something else”).
P2: If the universe is contingent, it must have (requires?) a cause or explanation.
P3: That “something else” is God, or the First Cause.
It seems to me that these are the typical arguments or objections made by various atheist scientists:
A1: While they might not reject the notion of contingency, they certainly don’t believe in the further claim of a “transcendent cause” (or explanation) of the universe. For example, in the Carroll-Craig debate, Dr Carroll said, “My claim is that if you had a perfect cosmological model that accounted for the data you would go home and declare yourself having been victorious.” You have a perfect model, case closed.
A2: They do not all reject the premise that a contingent being or fact has a cause or explanation, but deny any role for God.
From what I can survey, A1 rests on shaky methodological ground. I mention three arguments to justify this claim:
1. There is the 2π/7 problem discussed in Aron Wall’s blog post :
http://www.wall.org/~aron/blog/ where (a) the perfect model (truth) is elusive and (b) even the best model might still be unsatisfactory if understanding why a certain model parameter is exactly 2π/7 leads to new “puzzles and mysteries”.
2. William Craig argues for the Principle of Sufficient Reason (PSR) in defense of P2 in Argument for Contingency.
http://www.reasonablefaith.org/argument-from-contingency.
3. Luke Barnes argues in a blog post “Why Science cannot explain why anything exists”:
http://letterstonature.wordpress.com/2013/02/18/why-science-cannot-explain-why-anything-exists/
A: The state of physics at any time can be (roughly) summarised by three things.
1. A statement about what the fundamental constituents of physical reality are and what their properties are.
2. A set of mathematical equations describing how these entities change, move, interact and rearrange.
3. A compilation of experimental and observational data.
In short, the stuff, the laws and the data.
B: None of these, and no combination of these, can answer the question “why does anything at all exist?”
C: Thus physics cannot answer the question “why does anything at all exist?”
Similarly, we can go after Argument 2: rejection of God’s hand in creation. A refutation is in Luke Blog Post (see above):
1. Since the universe is the totality of space, time, matter and energy (i.e. that’s the sense of universe being used here), the cause of the universe must be spaceless, timeless, and immaterial.
2. The most plausible immaterial kind of thing that could cause a universe is a mind.
3. A spaceless, timeless, and immaterial mind that causes the universe deserves to be called God.
These are not proofs but arguments to justify the premises. St John Polkinghorne writes in Question of Truth, “But in most cases, question of truth, whether about science, religion, or any other field, are more elusive. There are seldom absolute knock-down arguments, one way or the other.”
But, are the arguments reasonable, plausible?
Final comment: Dr Aron, you wrote in your reply, “Maybe everything is necessary, and nothing causes anything else.” If true, is there no explanation for anything?
TY
Fascinating post. For reasons why one might exclude the infinite case, the you may find the following paper interesting:
Cohoe, C. (2013). There must be a First: Why Thomas Aquinas Rejects Infinite, Essentially Ordered, Causal Series. British Journal for the History of Philosophy 21, 838–856.
TY, thanks for your further thoughts. Two small comments:
1) "caused to exist by something else" is not the definition of contingent. The definition is something that might either exist or not exist. I think there is a reasonable intuition that anything in the latter category is also in the former category, but it doesn't follow by definition.
2) "Maybe everything is necessary, and nothing causes anything else". The universe might be deterministic, where everything hsa to exist for some reason. In that case we can still ask whether the reason appeals to something outside of that thing, and if so it would have an explanation. A cosmological argument might still be possible.
Another possible idea, that every single thing we see just exists necessarily with no other explanation seems like an extremely implausible idea, but my point was that one has to explicitly exclude this dumb idea with assumption P2, or the arguement can't prove a First Cause of other things.
Thanks for the link, Leon.
I've been pondering about the Leibnizian cosmological argument for a while, and I recently came up with an argument of why a quantum field(which atheist use against the argument) cannot be a necessary being and would be contingent.
The laws of quantum fields describe which arrangements of the fields are physically possible and which ones are not possible. If this is the case than that means than for a particular arrangement of a quantum field such as the one that gave rise to our universe (I'm not saying it is true, I'm just assuming it to show why if this were case it would not be necessary)there is a logical and mathematical possibility that it could have developed into a different arrangement of fields. If this is the case then that means that the current arrangement of the field that may have given rise to our universe could have been different and did not have to be the way they are. Since this is true, then it would fall under Leibniz's definition of a contingent being as it could have been different then the way that it is and did not have to be a specific way, even if it was eternal, and thus it would require ultimately a necessary explanation.
I'm not sure if this is valid as I don't know too much about quantum fields and I was wondering if you could help me see where I'm wrong in this thinking and how I could make it better,
Thank you in advance.
Kareem,
OK, let me help you translate this into physics lingo. Your basic intuition, that the laws of quantum field theory (as we known them) allow the universe to be in any one of multiple possible states, is indeed correct.
At any given moment of time, there is a state which describes the configuration of the fields. The laws of physics tell you which kinds of fields exist, and they also have a bunch of equations which the fields have to satisfy. The equations come in 2 kinds (a) constraints which place restrictions on the allowed configurations of fields at one moment of time (e.g. the Gauss Law in electromagnetism), and (b) equations of motion which say how to get from the state at some time
to some other time 
(or vice versa). (All of this is true whether we are talking about classical field theory or quantum field theory; it's not clear that the quantum mechanical nature of the fields is important to your argument, unless you wanted to make hay out of the fact that QM is a probabilistic theory, so that even once you've calculated the "state" you only have probabilities to measure various outcomes.)
For most reasonable choices of these laws of physics (including the Standard model of particle physics) there is more than one possible state (in fact infinitely many states) which are compatible with all the constraints at any given moment of time. Now, the laws of physics allow you to calculate what this state is if you know the "initial conditions". But the initial condition is itself arbitrary. Because the laws of physics allow many different states, it is in some sense physically "contingent" which state the universe actually is in. That is, not only is it logically possible for the universe to be in another state, it also would not violate any of the known laws of physics. (It is also logically possible for the laws of physics themselves to be different, but by definition it is not physically possible, since physically possible means compatible with the actual laws of physics.)
So that was just straightening out the physics part for you. To actually make the argument work you'd also need some metaphysical premises, basically some assumption about how things which are logically or physically contingent need some kind of cause.
An atheist who wanted to criticize your physics premises might argue that: (a) maybe there is some additional law of physics which determines the initial condition, such as the Hartle-Hawking no-boundary proposal. Or they might argue that (b) once we find the correct (unknown) theory of quantum gravity, it will turn out to have only one allowed state and no more. (Personally I would bet against these possibilities but I don't think they are totally crazy.)
An atheist who wanted to criticize your metaphysical premises might argue that a contingent thing can too exist without a cause, or that it doesn't make any sense for a necessary being to create a contingent being, because anything created by a necessary being would also be necessary. Or, they might argue that it is equally arbitrary to postulate a God having certain attributes when one could have postulated a different God having different attributes.
Well, I don't want to take away from you the fun of figuring out on your own how you'd like to respond to these objections... Good luck!
Thank you very much Dr. Wall, I''l get to working it out, and solving the possible objections
Dr. Wall,
Sorry to ask more questions, but there are some things which I would like to verify.
1. If a quantum field is eternal does that mean it has always been eternal in one state or it simply is eternal but is always changing it's configuration fields, and can always be different than it's current state?
2. Previously you stated that the laws of physics make it physically possible for the universe to be in an infinite number of states different than the ones that it is currently in. If eternal inflation is true, would the laws of physics still allow for it to be physically possible that for the current state a universe is in as a result of inflation to be different than this state, and since inflation had to begin at some point, would these laws also state that it is physically possible for it to not have begun?
Thank you in advance for your time.
Kareem,
1. Physicists don't usually say that a field is "eternal" so I'm not sure what statement I'm supposed to be clarifying. If you're asking what "eternal inflation" means, it's just a fancy way of saying that some parts of the universe continue inflating forever (because in those regions the "inflaton field" is excited). If you mean, are fields ever created or destroyed, the answer is that in the Standard Model, the number of fundamental fields never changes, but the state of the fields does change. (A field is just a name for a type of variable that depends on spacetime position.)
2. Yes, in the usual theories of inflation, there are only some particular states in which inflation occurs (typically rather special low-entropy states, which raises the question of why the universe was in such a state to begin with.)
Inflation has a tendency to wash out details of the initial conditions, thus a particular theory of inflation tends to drive the universe towards being in a particular kind of state (within the observable universe) when inflation ends. But there are also quantum fluctuations during inflation which produce an element of randomness in the initial conditions; in our universe this is what seeded the initial formation of galaxies.
3. If you're going to make a philosophical argument based on physics, it would be good to make sure you understand at least one physics system well enough to use the terms properly. So just to check, here's a pop quiz question for you (no one else is allowed to answer this question until you take a crack at it): Pretend you are Isaac Newton studying Celestial Mechanics (the theory of orbits in the solar system). What are the states, and what are the equations of motion? (Bonus question: are there any constraints?)
(For purposes of answering this question, you may treat the planets as point particles. This turns out to be okay because of something called Newton's theorem which says that the gravitational field outside a uniform sphere of matter is exactly the same as if all the mass were concentrated in the center.)
Sorry for the late response Dr. Wall, I was spending some time in Canda and I had little access to the internet there,
With regards to the equations of motions for celestial mechanics I believe that they are -GM divided by R, v squared over 2, v equals route -GM divided by R, and v squared over 2 - GM over R.
For the states I`m not sure what you mean by this, and I do believe that there are constrains for the equations.
PS: To clarify, I am not really trying to make an argument using only physics, my aim is to show that for anything described by the laws of physics such as an eternally existing quantum vacumm, it is possible due to being described by these laws for it to have been in a state different than what it currently exist in or to have had different intrinsic properties at a specific moment such as the creation of our universe. Anything described by physical laws based on what we know so far did not have to exist in the way in which it does, and could habe been different, thus rendering them contingent. This is what I mainly aim to show.
Kareem, thanks for responding to my challenge. Don't worry about the timing, I've been pretty busy as well.
But do worry about being as precise and accurate as possible in your reply. I can't make sense of your grammar; for example I don't know what the commas mean. Does "route" mean "square root?" Could you please write that out as an equation, using latex following the intructions here? (If you don't know latex, the basics are pretty easy and explained on the web in many places. If you muff up the equation, just post a correction and I can fix your original comment for you.)
I made sure to include equation support on my blog so that people could communicate precisely what they mean.
After writing the equation, please indicate what each of the variables means.
is Newton's constant 
, I'll give you that one for free, but you can't just say "M is a mass" and "R is a distance" without saying what object it is the mass of, or which points 
is the distance between. Making sure your expressions are completely unambiguous is an important philosophical discipline. Physics is good practice for this, since metaphysics is a thousand times slipperier. I'm worried that my original explanation was completely useless to you, since you didn't understand the terms in which I phrased it.
I understand you are trying to make a more general metaphysical argument here. But you can't just go around using terms like "state", borrowed from physics, unless you can indicate that you know what they mean in at least one context. No one will trust you with the metaphysics of quantum fields unless you first understand how classical mechanics works! "He that is faithful with little is faithful also with much, and he that is unjust with little is unjust also with much." (Luke 16:10)
So take a guess, and tell me: in the system consisting of the Sun, Mercury, Venus, Earth (and its moon if you wish), Mars, Jupiter, Saturn, Uranus, and Neptune, where each of these objects is thought of as a point particle, if I already know the laws of gravitation, what information do I need to specify about this system in order to predict, henceforth, the motion of each of these objects? That is the "state".
Oh, and at this point anyone listening in who is not Kareem (and also not a trained physicist like St. Scott Church) is also free to take a guess as well, about the answers to my questions about what are the states, equations of motion, and constraints (if any) in celestial mechanics. I'm not trying to harass Kareem in particular (sorry if it came across as too mean), I'm mostly trying to gain some insight into what kinds of explanations are useful and illuminating to people outside of physics.
Dr. Wall,
In your question, are we free to treat the force of gravitation between planets as negligible? In other words, can we assume that the sun is the sole source of the gravitational field we are considering?
Hi Matt,
Unfortunately, answering your question would spill some of the beans. The problem is this: For the system of planets/moons specified, we want to know where they are, where they've been, and where they'll be in the future. Given the laws of Newtonian gravitation (and perhaps more precisely, the classical mechanics framework in general), what parameters/values do we need to specify in order to calculate this explicitly?
One small hint... there are multiple ways to solve this problem, but the answer to Aron's question will be the same in all of them.
Good luck! :-) [Aron, if I've given anything away feel free to edit out the offending content.]
Matt,
Either way is fine. It's not like a class exam, where people are after grades rather than insight! So long as you pick one reasonable interpretation of my question and answer it, I'll be able to tell whether you understand or not. So you tell me.
Aron,
You asked,
If I understand you correctly, if the question is asking, given that we already know that any two objects with mass
and 
at a distance 
are gravitationally attracted towards each other's positions with a force proportional to 
, what other information must be specified to predict planetary motion, then it seems to me the answer is as follows:
We need (a) the initial positions and velocities of each body in the system, (b) the mass of the sun for
, (c) the mass of each planet for 
, (d) the length of the semi-major axis of each planet's orbit for 
, and (e) the equation
,where
. Did I miss anything?
[cleaned up the latex a little bit using the \text{} command, which gets rid of the italics and allows spaces, putting {} around the -11, and by using \times in place of a symbol it couldn't process which I assume was copied and pasted from elsewhere (I made the same mistake at first when I wrote my comment above)--AW]
Jack,
Close. You actually have too much information, rather than too little; the correct answer is simpler. This may have happened because the "r" in the force law doesn't mean what you think it does.
Also, you seem to have switched halfway through whether you wanted to answer St. Matt's question yes or no. In your first paragraph you say that
and 
can be any two objects, while in the second paragraph you reserve 
for the sun and 
for the planets.
Aron,
So I may have provided too much information too, but in a Newtonian formulation this is what I have. The state of the system consists of the positions and velocities of all of the objects in the system at some time t = 0. The equations of motion are found through
. For newtonian gravity, the force F is found to be 
where 
is the mass of a planet, 
is the mass of the sun, G is the gravitational constant, and r is the distance between the planet and the sun. In spherical coordinates, the acceleration of a planet is written as
The equations of motion then become
The positions and the velocities of the planets (i.e. the state of the system) are given by
and
respectively. Given
, 
and the equations of motion, it should be possible the specify the state of the system at any time t.
If this is right, would you mind if I work this out in the Lagrangian formulation of classical mechanics and post it to see if I've done that right as well?
Aron,
The sole reason for my mention of
and 
in that first paragraph was to reiterate the conditional element of your challenge: the hypothetical clause "if I already know the laws of gravitation, . . ." is the antecedent to your challenge question. Thus, I was simply specifying what it is that one knows, if one knows the laws of gravitation.
Then, in the following paragraph, I made my attempt at providing the answer to your challenge; and that is why I made explicit the substitution of variables at this point.
Despite the apparent appearances, my intent was never to switch anything with respect to answering the challenge. So I guess I'm not completely clear on what, exactly, the challenge is. Would you mind repeating it, please?
Aron,
I realized I wrote the position vector of each of the planets wrong. It should be merely
Matt,
That's basically correct, although there's no need to uglify your answer by writing it in a particular coordinate system, seeing as I don't expect you to calculate anything with it. (If you ever do need to do the calculation, it's a lot easier to use conservation of energy and angular momentum, then to use the Newtonian force law.)
Jack,
That was a minor point, it was just a minor confusion about what you were trying to do. My more major concern was that (a-c) is already a complete description of the state; (d) and (e) are redundant pieces of information which can be worked out from the orbit and the force law respectively.
The challenge was simply this: for the celestial mechanics system, to provide a complete description of the states and equations of motion. And, since St. Matt has provided the gist of the answer, I now reveal what I was looking for:
The STATE is simply the positions and velocities of all the celestial bodies.
.)
(Strictly speaking the masses are also part of the state, but for purposes of this problem it is more convenient to regard them as constants, like
The EQUATIONS OF MOTION are simply the Newtonian force law as applied to each celestial body, as written out by St. Matt.
(To make the problem more realistic, one could also include the gravitational effects of the planets on each other, but this can be calculated using the same equation, by simply adding up all the force vectors.)
And the answer to the bonus question is that there are no CONSTRAINTS, since every possible list of positions and velocities represents a physically possible state of the theory (except maybe putting two planets in the same place, but we are idealizing them as point particles so that's not a big issue.) But there would be constraints if we thought of the gravitational field as an independent object from the planets, because the planets determine the location of the gravitational field through Poisson's equation.
There is no constraint in the form of an upper bound on velocities (e.g.,
)?
Jack,
Not if you're doing Newtonian mechanics, which is nonrelativistic, so that the speed of light plays no role. But if the planets were to approach the speed of light, you should instead use Einstein's theory of GR, where there is a indeed a bound on the speed. GR is much more complicated, so the description of the states is different. (Since the gravitational field has independent degrees of freedom, the state would contain more information than just the positions and velocities of the planets.)
Gotcha. Thanks.
Just a suggestion: you should post some more questions/problems like this for everyone; seems like they would be both fun and educational. Again, just a suggestion, not me telling you how to run your blog. :-)
Jack,
Yeah, it was kind of fun; I had the same idea. If I can think of some more good theoretical physics questions, I might do so.
Awesome!
You guys might enjoy this: http://commonsenseatheism.com/wp-content/uploads/2009/05/craig-and-sinclair-the-kalam-cosmological-argument.pdf
Sorry--it's long, but pg 25 - pg 82 is a purely 'physics' approach to the Kalam argument. They basically try to rule out all eternal cosmologies that can't be subjugated by the BGV theorem.
[I removed this comment because it made no sense at all, and was posted to multiple blogs---AW]
I'm late but would like to comment.
Carrol sets up his particular conditions so he just knock them down when it comes to the universe and the - What gives us the right to demand a cause - is a nice touch of self righteousness so we can all feel so ashamed we could ever conjure up the novel and childish notion that the finely tuned, impossible object known as the universe didn't just pop into existence, complete with a set of laws that built order, stars, and then planets, which out of the chemicals and bonds,
a molecule emerges in the cart before the horse fashion, that contains the spacial location of a 4 chamber pump,
its voltage and electrical blueprints, tubing and complex pressure system. In other words it popped out People who are so clueless as to think there's something greater than us behind it all.
Since many atheists have these glaring inconsistencies
and favor a block universe, I've always thought a time lapsed picture should quiet anyone with a partially open mind.
Picture the building of reality sped up, seeing life and people just essentially popping out of the ground and still arguing that we have no right.
Luckily, most atheists escape this first stage of grief, denial, they are in. Last I looked less than 20% remain atheists. That alone can help some of them analyze their arguments, that really are a clinic of desperation, looking for technical reasons or loopholes, a patchwork of excuses that, to them, formulate an escape from what really is obvious to everyone.
I do like Craigs example of the translucent ball in the forest that demands a cause. Just because the universe is so big and inexplicable, atheists somehow reason it's OK for it to just pop out of nowhere uncaused or always exist.