Updated Study Questions and Reading Advice

There will be extra office hours from 3-4:30 in the lounge of the philosophy dept on Monday, May 12. The exam will be held at the normally scheduled time, Thursday May 15 at 9 AM in Wilson 303.

Sixth Assignment

You may choose either of the following two topics. The paper should be 1500 words. Again, target an educated audience that doesn't know anything about relativity. Assignment should be about 1500-2000 words. Assignment is due Monday May 12 at 5 PM.

Option One

Explain how best to interpret the relation between mass and energy. Try to make sense of Einstein's comment on page 48 of The Meaning of Relativity: "Mass and energy are therefore essentially alike; they are only different expressions for the same thing. The mass of a body is not a constant; it varies with changes in its energy." Consider the issues that arose in our reading for that topic. Try to clarify the extent to which the issues are not really clear.

Option Two

Explain to what extent Einstein vindicates Poincare's thesis that the nature of space is conventional, and to what extent does it vindicate an extreme empiricism like the logical positivists had? [Note that I did not write "spacetime." I'd like you to discuss the theoretical status of 3-dimensional space in relativity. You are free to discuss spacetime if you wish, of course, if that will help.] For some material, look at the beginning of Einstein's The Meaning of Relativity. For example,

In Relativity, p. 9, Einstein writes,

and on p. 10-11,

Examine what Einstein says in other parts of Relativity as well.

Final Exam Review

Here are a few things you should study for the exam:

• Read Friedman's introduction for the overview of the most important philosophical aspects of relativity.
• Read chapters 1-6 of Einstein's Relativity and think about its relation to operationalism, and the verifiability theory of meaning.
• Read the first 9 pages of Einstein's The Meaning of Relativity.
• Read 19.3-19.5 to review the hole argument.
• Read Chapter 8 of Brown's Physical Relativity.
• Read Lange's chapter on mass.
• Read Maudlin's paper on nonlocal correlations.
• Reread the notes except chapter 1. (Nothing technical on the exam.)
• You need to know the argument for why quantum non-locality is non-local causation Ch. 5 in Maudlin's book and be able to criticize it.
• On the hole argument, realize that there is a distinction between Einstein's hole argument and Earman's hole argument which he uses to attack manifold substantivalism. Don't worry about the complicated stuff in Earman's book about surgical holes in the spacetime manifold. Concentrate instead on the non-technical arguments about determinism, and the responses: metrical essentialism and counterpart theory. Can also read Norton on The Hole Argument.
• How is the classical principle of relativity different from the special principle of relativity?
• What (if anything) is wrong with saying the following? "It isn't just that quantum mechanics entails non-local correlations between some particles, but that the experimental evidence itself (the results of Aspect, et al.) that entails non-local correlations between some particles."
• How does the General Theory of Relativity affect the issue of non-locality?
• How does the General Theory of Relativity affect the application of the special principle of relativity?
• Why is it bad to explain the Special Principle of Relativity as a claim about the invariance or covariance of the laws of nature?
• What is the equivalence principle?
• How does the equivalence principle differ from the general principle of relativity?
• Why does the equivalence principle fail to "relativize" acceleration? (whereas the special principle of relativity does relativize velocities)
• Reread the notes below on the equivalence principles and principles of relativity.

Fifth Assignment

Your fifth assignment is to evaluate whether quantum non-locality violates relativity. You should explain what quantum non-locality is, and then compare various interpretations with regard to whether they conflict with relativity. You need to make clear various precisifications of what the theory of relativity is, in order to see what counts as violating relativity. You may want to take a look at chapter 1 of Maudlin's book as well as the other assignments we have read. Assignment should be about 1000-1500 words. Assignment is due Friday April 25 at 5 PM. Turn in a paper copy to me in class Thursday or in my office mailbox on the 1st floor of Gerard House (next to the List Art center.)

Fourth Assignment

Your fourth assignment is to criticize Einstein's argument in Appendix Five. You can be as sophisticated as you like in this paper, journal quality if you like. It would be a good idea to bring up the issues with the modern hole argument. Assignment should be about 1500-2000 words. Assignment is due Friday April 18 at 5 PM. Turn in a paper copy to me in class Thursday or in my office mailbox on the 1st floor of Gerard House (next to the List Art center.)

Third Assignment

Your third assignment is to explain the principle of relativity. (Note that this is NOT the THEORY of relativity, but the principle. See below.) You should distinguish it from relativity-principle-imposters (the textbook versions of the principle), pointing out what claims are trivial and what aren't. Include significant discussion of its vague points (like observability and isolation) in different contexts. (In EM, what is observable, in GR what does isolated mean) Discuss how Einstein's principle of relativity is different from the classical (Galilean) principle of relativity (if you think it is). The target audience is just like in the first paper, an educated audience that doesn't know anything about relativity. Assignment should be about 2000-2500 words. Assignment is due Wednesday March 19 at 5 PM. Turn in a paper copy to me in class Tuesday or in my office mailbox on the 1st floor of Gerard House (next to the List Art center.) In emergencies, you can email the paper to me by 5PM, but if you do so, you need to turn in a paper copy the next chance you get. The electronic copy is only to prove that you had your work done in time.

The Principles of Relativity

It is very important that you don't confuse the principle of relativity with the theory of relativity. These are not the same thing.

The special principle of relativity makes the claim that physical systems that are related by a shift, static rotation or boost are indistinguishable (from within the system). This principle is also expressed by saying that inertial systems are physically equivalent (indistinguishable by any mechanical experiment). Sometimes it is phrased as a claim that the laws of nature hold in all inertial frames (as opposed to just holding in a rest frame of the universe if there is such a thing). [I am not strictly speaking endorsing all of these as fully accurate expositions of the special principle; I am just reporting different ways the principle has been expressed.]

The special theory of relativity is the physical theory that Einstein arrives at by using the special principle of relativity and the law of light to derive the Lorentz transformations. It supports electromagnetism in the sense that electromagnetism can be formulated within the constraints of special relativity in a way that makes the speed of light a constant (relative to any particle whatsoever), and makes electromagnetic phenomena depend only on the relative positions and relative velocities of particles.

The general principle of relativity is a bit more shifty. Einstein thought of it as a generalization of the special principle to motions that are non-inertial. A different conception of it that appears in Einstein's writings is that it is a Machianization of absolute acceleration. Yet another principle that Einstein thought was closely related (or equivalent?) was general covariance.

The general theory of relativity is the physical theory that extends the special theory of relativity by dropping the requirement that the connection is flat and replacing it with Einstein's field equations, G=kT, and construing the gravitational force as merely encoded in the curvature of spacetime so that there really is no gravitational force.

Your paper assignment is to explain the principle of relativity, not the full corresponding theories.

Second Assignment

Your second assignment is to explain how to resolve the twin paradox in a cylindrical spacetime. See the Lockwood excerpt for a non-technical setup of the problem. An explanation is given by Dray, but it isn't explained very well. I'd like you to explain what's going on in less technical detail and with much greater clarity. The version I wrote as a sample answer is only about 700 words long with 4 diagrams. That should do, but make sure you thoroughly tease out the consequences for how the example clarifies the lessons of special relativity. See if you can clarify the philosophical significance of this version of the paradox. I think a very useful tool to help figure out what is going on, is to remember that the surface of a cylinder is topologically equivalent to a rectangle with one pair of opposite edges being identified with one another. Just draw a graph of the spacetime points and calculate what happens under a Lorentz boost.

Assignment due Friday March 1 at 5 PM. Turn in a paper copy to me in class Thursday or in my office mailbox on the 1st floor of Gerard House (next to the List Art center.) In emergencies, you can email the paper to me by 5PM, but if you do so, you need to turn in a paper copy the next chance you get. The electronic copy is only to prove that you had your work done in time.

Relativization of Concepts

What makes the Theory of Relativity deserve its name is that certain concepts which have a frame-independent meaning in Galilean spacetime/Classical mechanics, have a frame-dependent meaning in Minkowski spacetime/relativistic mechanics.

The most central concept that becomes thus "relativized" is simultaneity. In Galilean spacetime, there is an objective fact about whether any two given spacetime points are simultaneous. In Minkowski spacetime, two spacetime points A and B are simultaneous if A=B, and non-simultaneous if they are time-like separated. If A and B are space-like separated, there is no fact of the matter about whether they are simultaneous. To the extent that one can even meaningfully say that there is a fact about their simultaneity, it is established by convention. One just stipulates by arbitrary convention a slicing of the spacetime into space-like slices.

Another relativized concept is spatial length. Length in Galilean spacetime is the spacetime distance between two simultaneous spacetime points. Length in Minkowski spacetime is also the spacetime distance between two simultaneous spacetime points. But because simultaneity is at best conventional, so is length.

For an object that is inertial, there is an objective fact about what its length is in its rest frame. This is because there is an objective fact about which points are located in the spacetime slice that is orthogonal to the object's world-tube. This length is called the proper length. In any inertial frame boosted with respect to the rest frame, the length of the object (assuming the convention that spacetime is sliced into hypersurfaces orthogonal to the world-lines of particles at rest in that frame) is smaller than the proper length. This is called length contraction.

Another relativized concept is the concept of 'being contained inside'. What it means for an object A to be contained inside a container B is that all of A's parts have to be inside B at the same time. Because 'at the same time' is at best conventional, in some cases it turns out to be conventional whether A is inside B.

Another relativized concept is rigidity. What it means for an object A to be perfectly rigid is for A's parts to maintain their spatial distances over time. This means that if one part of A is accelerating, all other parts of A must accelerate equally at the same time. Because 'at the same time' is at best conventional, in some cases it turns out to be conventional whether A is behaving rigidly.

A concept that is not relativized (to a reference frame) is temporal duration. The amount of time that passes for a particle is the length of its worldline. Duration is path-relative, it depends on the object's path through spacetime. Duration is not frame-relative because the length of a world-line is an invariant quantity, depending only on intrinsic features of the world, i.e. the object's path and the Minkowski metric.

First Assignment

Explain (to a popular audience that is well educated but has no sophisticated understanding of physics) how acceleration can be absolute while velocity is relative. (It helps as part of your explanation to explain several arguments for the absoluteness or relational character of each quantity.) Assignment should be about 1500-2500 words, and diagrams will help. I will be grading the material not only on whether it is correct, but whether you are able to communicate the concepts effectively to someone who doesn't know any differential geometry. Assignment due Friday Feb 14 at 5 PM. Turn in a paper copy to me in class Thursday or in my office mailbox on the 1st floor of Gerard House (next to the List Art center.) In emergencies, you can email the paper to me by 5PM, but if you do so, you need to turn in a paper copy the next chance you get. The electronic copy is only to prove that you had your work done in time.

The Absolutist-Realist Debate

Here is an overly simplistic version of the debate to serve as a good first approximation of the different sides of the issue:

Relationists (Liebniz, Mach) argue that space is an idealization that is grounded merely in the distance relations between material objects. Space doesn't really exist. Fundamentally, the universe consists of different clumps of matter with distances between them. The distances are fundamental relation facts and are not derived from facts about where objects are located. It is as if God's inventory of the universe consists of a spreadsheet with all the objects listed vertically and horizontally, with a triangular patch of numbers indicating the distances between them. The only sense in which space exists is that we imagine that these distance relations are actually imbedded in three dimensions.

Absolutists (Newton, Clarke) argue that space is an entity in its own right, that space would exist even if there were no matter in the universe. The structure (properties) of space means it makes sense to talk about quantities like absolute position and absolute velocity, even though we have no epistemological access to the absolute position or velocity of anthing in the universe. The classic justification for believing in absolute space is that a sufficient explanation of centrifugal effects in rotation requires more than just relational quantities. Toy version of Newton's bucket argument: The still bucket and rotating bucket are relationally equivalent. They are physically different. Thus, space is not relational. Thus, space is absolute (Newtonian).

Newton's Laws of Motion

Here are the relevant principles of Newtonian Mechanics as Newton wrote them in the Principia:

Law I: Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.

Law II: The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

Law III:To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

Also relevant is the inverse square law of gravity, (Book III, Proposition 7, Theorem 7): "That there is a power of gravity pertaining to all bodies, proportional to the several quantities of matter which they contain." And a few paragraphs later in Corollary II: "The force of gravity towards the several equal particles of any body is inversely as the square of the distance of places from the particles;... "

Newton's Rules for Reasoning in Philosophy

Rule I: We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.

To this purpose the philosophers say that Nature does nothing in vain, and more is in vain when less will serve; for Nature is pleased with simplicity, and affects not the pomp of superfluous causes.

Rule II: Therefore to the same natural effects we must, as far as possible, assign the same causes.

Rule III: The qualities of bodies, which admit neither intensification nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the unversal qualities of all bodies whatsoever.

Rule IV: In experimental philosophy, we are to look upon propositions inferred by general induction from phenomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phenomena occur by which they may either be made more accurate, or liable to exceptions.

Einstein on Classical Mechanics and the Principle of Relativity

In chapter 3 of Relativity, Einstein explains how he thinks we should understand the concepts of space and time in classical mechanics:

Einstein here is endorsing a version of relationism that in order for there to be motion there needs to be a physical body with respect to which the motion is relative. Einstein then goes on to distinguish a mathematical device, a system of coordinates "rigidly attached" to the body of reference.

Some interpretation is required to understand what "rigidly attached" means. I think Einstein intends a coordinate system adapted to an inertially moving body. That is, if there is a body that is under the influence of no net forces, we can label the points of spacetime using one axis that coincides with the spacetime path of the reference body. We can call this axis the time-related axis. From that point, one would like to argue that we can arbitrarily pick three other space-related axes that are perpendicular to each other and to the time-related axis, and that we can extend these axes in a straight line out to infinity. We can indeed to this, but for perpendicularity and straight-line to make sense, spacetime must have a metric (or something rather close to a metric). Here is where Einstein appears to be sneaking in spacetime structure that is not being tied to material bodies. That is, Einstein's relationism requires that the structures of spacetime be tied to material entities, but in this quoted passage, at least, Einstein is implictly relying on the fact that there is a spacetime metric, and that a corresponding spacetime connection everywhere flat.

Mach

I've been reading an excerpt from Ernst Mach's Die Mechanik in ihre Entwicklung. In the translation I am reading, the title of the excerpt is "Newton's Views on Time, Space and Motion." It is interesting historically to see how he continually misses the relevant point of Newton's bucket argument. Repeatedly, Mach claims that the only relevant motions needed to do physics are relational, but never backs this up with a real argument. In one excerpt, he argues

Notably, Mach correctly notes that we are able to notice not only a relative acceleration between K and K', but an absolute acceleration defined for K and K' separately. But after admiting that the acceleration is not merely relative (at least to K and K'), he claims that this acceleration is only detectible because there are other bodies present. But this is incorrect because one can calculate how quickly the distance between the bodies should change, given that they are not revolving around each other. One can measure empirically how quickly they are approaching one another, and then calculate how quickly K and K' were rotating. Of course the velocity of K and K' with respect to absolute space cannot be measured, as Newton says, but even total knowledge of the relative distance and velocity of K and K' in conjunction with the knowledge of the accelerations of each body is insufficient to predict the future motion. Adding external bodies A, B, and C to space does not help to make the prediction. It seems Mach is confusing the issue of what kinds of facts are available to us empirically, like relative velocities, and what kinds of facts we need to postulate in order to explain and predict motions, e.g. absolute accelerations.

Here is another comment Mach makes, immediately after quoting Newton's bucket argument. See if you can analyze what is wrong with it.

Explanation of The Special Principle

Remember that the Special Principle of Relativity cannot be captured by the claim that "the laws of nature take the same form in all inertial reference frames" or that "the laws of nature are the same (invariant) in all inertial reference frames." This is because the laws of nature describe structural relationships between physical quantities. These relationships are objective and are thus independent of how we choose to label the parts of space and time. The laws of nature are true full stop. Hence, they are true in all reference frames. Hence, they are true in all inertial reference frames. All of this is so, regardless of the content of the theory.

The intuition behind the special principle is that if a chunk K of the physical world is sufficiently isolated from the external environment, then the physical goings-on in K are independent of the position, orientation, and velocity of K. The claim of the special principle can be broken into two parts. First, assume it is possible to have an isolated chunk of physics K that contains experimenters who can measure quantities of the things in K, and let K' be any exact physical duplicate of K except for the fact that K' is located somewhere else and K' might be oriented in a different direction than K and might be moving relative to K. The special principle of relativity claims that any experimenters in K doing measurements of stuff in K will always get exactly the same results as experimenters in K' doing measurements of stuff in K'. The claim is that K and K' are (internally) indistinguishable in all respects.

Second, the principle of relativity implicitly involves the claim that if K and K' are indistinguishable in all respects, then we should treat them as theoretically equivalent. That is, we should never admit the existence of any physical structure that has as a consequence that the objects and events in K are in some way different from those in K'.

For example, Einstein (correctly) took it to be a law that the vaccuum speed of light (relative to every slower-than light object) is a universal constant c. This motivates us to seek a theoretical structure that accommodates this fact while only distinguishing relative velocities. After some research, we find that Minkowski spacetime serves these purposes if we interpret light as a particle that always follows paths of length 0.

The special principle, thus explicated, has some difficulties. First, if the laws of evolution are indeterministic, the experiments in K and K' might show different results just because the chance-outcomes are sometimes different on different trials and not because there is a meaningful absolute velocity. Second, the notions of indistinguishability and isolation are loaded concepts.

For example, what is indistinguishable depends on one's capacity for detection and on what one takes to be detectible. It is possible that there are limitations on what we humans can detect experimentally such that our experiments might lead us to infer that there are no absolute velocities in cases where an omniscient being could "see" that absolute velocities play a role in the evolution of the physics. This situation could arise in some relativistic quantum-mechanical treatment of collapse. A theory might posit some preferred global slicing of the spacetime into planes of simultaneity, and thus ground absolute velocities, just in order to have well-defined wave-collapses, yet constrains our ability to precisely measure particles in a way that prevents us from determining which slicing of spacetime is the preferred one. It could also arise in a relativistic extension to Bohm's deterministic interpretation of quantum mechanics, where a preferred slicing might be needed to make sense the dynamical laws even though the slicing is epistemologically inaccessible to us.

It should also be noted that future discoveries of what kinds of quantities are empirically distinguishable might throw into doubt whether K and K' are distinguishable. It is always assumed in explications of special relativity that the electromagnetic field quantities are not observable except by their eventual consequences for material particles. For example, there are non-trivial rules for how the electric and magnetic field quantities transform under Lorentz boosts. If we could observe the strength of the electric field, E, then we could distinguish K and K'. It turns out to be a reasonable assumption that we cannot observe E except by judging its effects on the motion of particles. Beautifully, the relative motions of particles in the presence of electromagnetic fields is preserved under Lorentz transformations, even though E itself changes, so that the values for the electric and magnetic fields are definable only against a conventional choice of frame. Furthermore, in general relativity, the fields can also affect the motions of particles by contributing to the spacetime curvature, but in this case too, the affect on spacetime comes not by way of E alone but by way of the Lorentz-invariant stress energy tensor. Nevertheless, it is worth considerable thought, on what grounds we take only particle positions to be our epistemological starting point for determining what is empirically distinguishable.

The Equivalence Principle

Einstein describes the following thought experiment:

Then Einstein continues by drawing the conclusion that the inertia of an object must equal its gravitational (coupling) mass.

Although Einstein expresses the connection between inertial mass and gravitational mass as equivalence, in fact all we need is proportionality. This is because any factor relating them can be folded into Newton's constant, which is possible because, so far as we know, Newton's constant cannot be derived in terms of other independently measurable quantities.

The expression "equivalence principle" is used variously for several different claims: (1) Linearly accelerating is physically equivalent to being at rest in a gravitational field. (2) The inertial mass and gravitational coupling mass are proportional to one another in all physical bodies. (3) "In any and every local Lorentz frame, all the (non-gravitational) laws of physics must take on their familiar special-relativistic forms." (Misner, Thorne, and Wheeler; p. 1060) and (4) the spacetime connection is unique (Friedman, 1983).

There are a couple of senses in which (1) is true. Because in GR absolute acceleration is just the deviation from the straight-line motion determined by the connection, standing on earth for example is literally being accelerated upward by the ground at 9.8 m/sec^2. Also, (1) seems to hint that there is a degree of freedom in the theoretical description of a situation that makes facts about absolute acceleration depend on choices about what the true gravitational field is, or vice versa. This is true in the curved spacetime version of Newtonian gravitation. Without special physical situations and boundary conditions, there is no unique solution for the gravitational field. For any gravitational field G, the field G+df will satisfy Poison's equation too. There is also a corresponding freedom in the affine connection that can balance out the df part. Thus, in the standard classical mechanics with gravitation, we have no way of distinguishing physically linear accelerations from global gravitational fields that aren't tied to any material body.

However, as Friedman argues, the indistinguishability of linearly accelerating frames in classical mechanics with gravitation does not motivate a successful relativization of linear acceleration. His argument is that when we modify classical mechanics with gravitational field to accommodate the indistinguishability of linear accelerations, we do this by claiming that there is no gravitational field, and instead that there is an affine connection that satisfies the appropriate gravitation-like relationship to the source masses. In so doing, the symmetry group of this curved-spacetime classical mechanics becomes the group of all rigid Euclidean motions. This includes linear rotations but even more, it contains rotations as well. (Remember that the symmetry group is the mathematical group that preserves the component-values of non-dynamical elements of the theory. Because the affine connection is tied to the mass distribution, it is now dynamical, and so the transformations only need to preserve the Euclidean metric of space and time separately. Hence, it contains arbitrary rotations.) Nevertheless, the indistinguishability group of the curved spacetime theory is NOT the linear accelerated motions, but the much smaller group of local Galilean boosts and static rotations. For example, a shift in space is in general distinguishable, because the matter might be shifted to a spatial location where the affine connection is different from the original location. (Here, we are assuming that the right way to interpret a shift is to hold fixed the geometry determined by the external masses. This is questionable. See the comments on the special principle of relativity.)

Special Principle of Relativity

Friedman argues that we should understand the special principle of relativity as the conjunction of two claims.

• R1: All inertial frames are physically equivalent. Physical equivalence can be understood as indistinguishability by any mechanical experiment.
• R2: If two models are physically equivalent, they should be theoretically equivalent.

The way this works in practice to motivate Minkowski spacetime as the sole spacetime structure, according to Friedman is that we notice that theoretically, Maxwell's equations + the Lorentz force law only can hold in rest systems, but experimentally, all electrodynamic phenomena are indistinguishable under boosts. Thus, our theory is wrong. We notice that Maxwell's equations can be interpreted as being covariant under boosts if we understand the boosts as Lorentz boosts, not Galilean boosts. So the problem is the Lorentz force law, so we change it to make it Lorentz invariant, thus satisfying R1. Then we notice that the corrected version of the force law is still formulated in terms of undetectable structures like velocity field (that represents the velocity of points of absolute space). So because of R2 we drop these undetectable structures, and we end up with relativistic electromagnetism.

The General Principle of Relativity

One of the motivations for generalizing the relativity principle is that the special principle of relativity seems to give special significance to inertial motions (inertial frames). Einstein thought this inadequate. There were several issues that were worrying:

The Equivalence Principle

One special kind of non-inertial motion, uniform linear acceleration, is noted to be locally indistinguishable from an inertial motion in the presence of a uniform gravitational field:

The relevance of this example is that it highlights a pair of theoretically different situations that are (allegedly) indistinguishable. So it opens up the possibility that a more general principle of relativity could make it such that these two indistinguishable situations can be understood to be theoretically identical.

The unobservability of the affine connection

The inertial structure of spacetime affects the motion of matter but is not observable.

Unfortunately, Einstein often expressed the general principle of relativity as a matter of the laws holding true for all coordinate systems, i.e. being generally covariant. It is trivially true that all laws concerning spacetime are generally covariant.

Review for Mid-Term Exam

Newton's Scholium to the Eighth definition. What claims is he making. Earman, I think, has a good gloss on Newton's view.

Leibniz's arguments from Identity of Indiscernibles and Principle of Sufficient Reason. The most important quotations are these:

Also for good measure is Leibniz's response to the bucket argument:

I find nothing in the Eighth Definition of the Mathematical Principles of Nature, nor in the Scholium belonging to it, that proved, or can prove, the reality of space in itself. However, I grant there is a difference between an absolute true motion of a body, and a mere relative change of its situation with respect to another body. For when the immediate cause of the change is in the body, that body is truly in motion; and then the situation of other bodies, with respect to it, will be changed consequently, though the cause of the change be not in them. 'Tis true that, exactly speaking, there is not any one body, that is perfectly and entirely at rest; but we frame an abstract notion of rest, by considering the thing mathematically. Thus have I left nothing unanswered, of what has been alleged for the absolute reality of space. And I have demonstrated the falsehood of that reality, by a fundamental principle, one of the most certain both in reason and experience; against which, no exception or instance can be alleged. Upon the whole, one may judge from what has been said that I ought not to admit a moveable universe; nor any place out of the material universe.

Read Huygen's thought experiment in Earman, Chapter 4, section 3. See if you can come to some conclusion here. (Be able to explain and criticize Huygen's argument here.)

Think about the quote from Leibniz that appears in Earman, Chapter 4, section 4:

Review Mach's response to the bucket argument in Earman, Chapter 4, section 8.

You need to be able to know what spacetime and physical structures are needed for classical physics and relativistic electrodynamics.

Why is Galilean spacetime the appropriate spacetime for classical mechanics and Minkowski spacetime the appropriate spacetime for electrodynamics?

There will not be a heavy concentration on the mathematical devices that we spent 2 weeks covering. But you should still understand why the devices we needed were created. E.g., a manifold was needed so we could take derivatives of scalars, a connection so that we could take derivatives of vectors, a metric so that we can measure distances and angles, etc.

You need to be able to do some basic conversion of points from one reference frame to another using the Lorentz boost transformation.

Understand in what sense Classical Electromagnetism (in Newtonian spacetime) constitutes a success for Newton's programme. In what sense did Electromagnetism prove to be the downfall of Newton's programme?

In what ways does the move to Minkowski spacetime affect the relationist absolutist debate?

Einstein used an argument that there ought to be a unified explanation of how a magnet causes a wire to conduct electricity regardless of whether the magnet is moving towards the wire or the wire is moving towards the magnet. Velocities, after all, ought to be merely relational. Norton argued that Einstein could have accommodated a fully relational account of the conductor-magnet experiment using just Galilean spacetime. That is, the conductor-magnet explanation problem does not necessitate special relativity. What is wrong with just using Galilean spacetime to explain the conductor-magnet phenomena?

How is length different from time insofar as they are grounded in relativistic physics?

How does rotation pose a problem for relationism that is different from the problem posed by linear accelerations?

Think about Einstein's explanation of the principle of relativity.

Know the difference between covariance and invariance, and how those two concepts relate to the principle of relativity.

Sample Answers to Mid-Term Exam

1. Huygens' argument is that the true motion of the objects (i.e., absolute velocity) cannot be determined before the balls strike the rod. After they strike the rod, the relative motion of the two balls is still oppositely directed, the only difference being that the direction rotates. Thus, the true motion of the objects still cannot be determined, and thus the true motion is indiscernible and hence metaphysically superfluous.

There are several problems with the argument. For one, the notion of true motion is ambiguous. Huygens' premise assumes that true motion means absolute velocity, which correctly is still epistemologically inaccessible after the balls have stuck. There is still absolute acceleration though, which is a kind of absolute motion.

For another, Huygens treats the rotational motion as if the rotating direction of the velocities can be ignored. They can't because they are needed for the phenomena of rotation.

For a third, Huygens is cheating by using the relative velocity of the balls after they stick in order to say that the motion is directed in "opposite parts of the circumference." The relative velocity of the balls is zero, and hence is not directed anywhere.

2. Leibniz used (1) the Identity of Indiscernibles (II), which is the principle that if two things are completely indistinguishable from each other, then they are literally identical and (2) the Principle of Sufficient Reason (PSR), that everything that happens or is true, happens or is true for a (good, non-arbitrary) reason.

Leibniz thought these two principles effectively implied the same consequences. However, these two principles can be made to come apart if we read II as relying on a notion of 'indiscernible for human beings,' as Leibniz seems to intend in his arguments about space. Then, the PSR could permit the use of superfluous spacetime structure if God can have access to facts that are inaccessible to human beings.

3. Galilean spacetime is appropriate to Newtonian physics in the sense that (1) it contains all the necessary structure to make sense of Newton's three laws of motion and his law of gravitation and (2) it contains no superfluous structure that would create matters of fact which we have no way to discover given Newtonian mechanics. For example, any two worlds related by a translation, static rotation or boost are indistinguishable by the Newtonian laws and are indistinguishable according to the Galilean spacetime structure.

The Galilean transformations are mappings of the spacetime structures representing translations, static rotations, and boosts. They are symmetries in the sense that no matter what coordinate system one uses, all the components of the spacetime structures (metric, connection) remain unchanged under a Galilean transformation. Specifically, Newton's law and the law of gravitation written in their traditional form where the connection components are set equal to zero, are invariant under exactly the Galilean transformations.

Minkowski spacetime is appropriate to electromagnetism in the sense that (1) it contains all the necessary structure to make sense of Maxwell's laws and the Lorentz force law and (2) it contains no superfluous structure that would create matters of fact which we have no way to discover given the laws of electromagnetism. For example, any two worlds related by a translation, static rotation or boost less than the speed of light are indistinguishable by the laws of electromagnetism and are indistinguishable according to the Minkowski spacetime structure.

The Lorentz transformations are also mappings of the spacetime structures representing translations, static rotations, and boosts less than the speed of light. They are symmetries in the sense that no matter what coordinate system one uses, all the components of the spacetime structures (metric, connection) remain unchanged under a Lorentz transformation. Specifically, the Maxwell equations and Lorentz force law written in their traditional form where the connection components are set equal to zero, are invariant under exactly the Lorentz transformations.

5. Laws express facts about physical structures. Hence it is trivial that the laws of electromagnetism and mechanics hold true in the same frames. They hold true in all frames! He could have characterized the principle of relativity as saying that electromagnetic phenomena (just like mechanical phenomena) are physically unaffected by translations, static rotations and boosts. Or he could have said any world is indistinguishable from a translated/rotated/boosted version of that world by any experiment.

Sample Answers to Sample Final Exam

(Not all answers completed.)

1. Here is an explanation of the equivalence principle in Symmetries, Lie Algebras, and Representations by Jürgen Fuchs and Christof Schweigert:

Critically examine the claims.

2. Misner, Thorne, and Wheeler explain Einstein's Equivalence Principle as a rule for determining how generic laws of nature that hold in special relativity can be modified to apply in general relativity:

Critically examine this claim.

3. "Explain and criticize Earman's version of the hole argument.": The argument tries to show that manifold substantivalism fails because it implies that determinism is false regardless of the laws of nature, whereas one ought to have metaphysical positions that make the truth of determinism hinge on physics.

4. "Explain what difficulties arise when one tries to apply the special principle of relativity in the context of the general theory of relativity.": The special principle of relativity is the claim that an isolated system will give all the same mechanical results (e.g. distances between particles) for an experiment conducted wholly inside the system, regardless of whether the system is shifted in space or time, oriented differently, or boosted. The biggest problem in applying this principle in GTR is that the condition that the system be isolated implies that the system be isolated from the gravitational effects of external masses. But, since in GTR gravitational interactions do not come by way of a force, but are incorporated through the metric, one must impose the no-interaction constraint through imposing a constraint on the spacetime structure. In general there can be more than one solution for the metric in the absense of any matter or fields (e.g. gravitational waves), and one's choice of such solution for the background metric will make an observable difference in the motion of matter.

5. "What is Maudlin's argument that the Bell correlations constitute an example of genuine non-local causation?": First, we assume that the event of the left photon passing its polarizer and the event of the right photon passing its polarizer are distinct events. Next, we notice that if the polarizer settings are the same on both wings of the (Aspect et al. type of) experiment, then when the two polarizers are aligned and both photons pass through, then we can say that by law if the left photon had not passed the left polarizer, then the right photon would not have passed the right polarizer. Next, we say that the distinct events A and B are causally implicated if: had A not occurred, B wouldn't have occurred. Then, we take causal implication as involving either A causes B or B causes A or there is some other event C that causes both A and B. Bell's argument rules out any possible common cause C that lies in the shared past light cone of both detection events. Thus, either one of the photons superluminally causes the other to pass, or there is a common cause C that superluminally causes both to pass. The possibility of common causes from the future are set aside for the purposes of this argument; they could allow purely local causation to explain the correlations.

6. "Why can't superluminal particles explain the Bell correlations?": In order for tachyons to explain the correlations, the type (or existence) of a tachyon sent from the left wing of the experiment must communicate the left-polarizer setting to the photon in the right wing of the experiment before it enters the right-polarizer. But this very fact, when reinterpreted in terms of certain other inertial frames, means that the photon on the right sends a tachyon to the left that correlates with the left-polarizer setting. But this requires the kind of non-local correlation that we are seeking to explain.