Fifth Assignment

Your fifth assignment is to explain to a lay audience the sense in which relativity and quantum mechanics are in tension with one another. You should explain what quantum non-locality is, and then describe various interpretations and discuss the extent to which 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 want to explain why the conflict is not outright contradiction, but that quantum mechanics may appear to flout the spirit of relativity.

The target audience is just like in the earlier papers, an educated audience that doesn't know the details about relativity.

Assignment should be about 2000-2500 words. Assignment is due Wednesday April 20 at 5 PM. Turn in a paper copy to me in class or in my office mailbox on the 1st floor of Gerard House (next to the List Art center.)

Mid-Term Exam Review

Read Einstein's "Appendix 5: Relativity and the Problem of Space" and think about your reactions to his various claims.

Also, review the alternative definitions of the equivalence principle.

Some questions to study:

1. Comment on the following definition of special relativity: Special relativity is the claim that all laws of nature are Lorentz-invariant.
2. Explain the status of spatial distance in general relativity.
3. To what extent does the debate over substantivalism dissipate if one's theory of matter is a plenum, a field quantity that inhabits all of spacetime?
4. To what extent does general relativity fulfill Mach's goals.
5. What is the difference between Machianization and Relativization?
6. Why do objects weigh more when they become hotter?
7. What is Einstein's equivalence principle?
8. What is the relationship between Einstein's equivalence principle and the strict proportionality of the inertial mass and the gravitational coupling mass?
9. Explain the (modern) hole argument. What are some responses to it?
10. What is different about Leibniz's arguments against substantivalism when they are posed in terms of hypothetical spatial shifts of all the matter in the universe versus hypothetical boosts of all the matter in the universe?
11. How good is the case for relationism in the context of special relativity where the relations are spatiotemporal distances?

Fourth Assignment

Critically analyze John Norton's The Hole Argument. The target audience for this paper is a professional audience who is familiar with basics of the hole argument. The assignment should be about 1000-1500 words. Assignment is due Wednesday April 6 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.

Course Description

The goal of this course is to analyze a wide range of philosophical issues that are informed by relativistic physics. The primary issue will be the philosophical debates over the ontological status of space and time, starting with the dispute between Newton and Leibniz about whether space is something above and beyond the properties of material bodies, and tracing how this debate evolved as attention shifted to relativistic theories. This debate was a motivating philosophical consideration in Einstein’s development of relativity, so we will examine Einstein’s role in the debate and how logical positivists interpreted relativity as supporting their broader philosophical program. Also, we will examine how relativity bears on our understanding of matter and energy.

There are no formal prerequisites for this course, but mathematics will be used in the course. I will presuppose that you understand differential calculus, vectors, and matrices. We will discuss the necessary elements of differential geometry for understanding the conceptual problems in relativity.

Your grade for the course will be determined by these factors:

1. There will be six homework assignments throughout the semester, worth a total of 60% of your final grade. The goal of these assignments will be for you to write expositions of theoretical and philosophical issues that arise in relativity. You will be attempting to write articles of the kind a sophisticated science journalist would write, pitched at an educated adult audience. Two of the assignments will involve solving some simple physics problems.

2. There is a take-home mid-term exam worth 15% of your final grade. It will involve criticism of existing textbook and popular presentations of various aspects of special relativity, as well as analysis about the range of philosophical interpretations of the theory of relativity and spacetime physics more generally.

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.

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.

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.

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 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.) 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.

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 4 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.

Third Assignment

Your third assignment is to explain the principle of relativity. (Note that this is NOT the THEORY of relativity, but the principle.) 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 Friday March 18 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.