crying 2

the usual view on NM/TRI (Albert)

1. the world is point particles; only whose positions are dynamical variables

2. the whole law of motion is described by F = ma

3. determinism: given all positions, velocities, and particle types, the whole history/future follows with certainty

4. TRI proof: x(t) to x(-t) is fully legal with no effect on acceleration

// therefore, no lawlife assymetry between past and present though experience shows asymmetry in physical processes, epistemic access and intervention

Hutchinson: NM is not TRI

begins by introducing a conceptual distinction between reversible vs. irreversible processes

1. SHM: ideal spring (reversible) vs damped oscillator (irreversible)

2. both are routine, clasiscal mechanics

3. the question of reversible vs irreversible has to do with geometric configurations vs asymmetry in time/velocity

// therefore, NM is neutral on reversibility. NM being reversible does not allude all systems are reversible. conflating theoretical reversibility for nature.

Hutchinson replies

microstructure: irreversible forces are phenomenal and able to reduce to reversible microforces

- this theory is untestable in nature; modern views see all forces as phenomenal

Greene's problem with the direction of time

1. everyday processes occur in one temporal order

(example) eggs splattering, ice melting, memories of the past

2. laws of physics are time-reversal symmetric: v(-t) is valid and simply makes a system retrace its path

3. laws permit reversible processes and don't distinguish past from future

// the arrow of time cannot be read from our laws

Thermodynamics: entropy

1. entropy = number of microscopic arrangements given the configuration of the larger picture (high entropy = many arrangements)

2. one correct order vs increasingly many disordered states based on number of constituents

3. 2nd law of thermodynamics: systems tend to evolve toward higher entropy

4. the second law of thermodynamics is not a fundamental law of nature, but rather, probabilistic/statistical reasoning applied to a time-symmetric laws

// therefore, entropy provides an arrow to time in systems with many consituents

Statistical mechanics

1. A microstate specifies every constituent's exact position and velocity. A macrostate is the gross description (ordered or not, spread out or not).

2. Many microstates realize one macrostate, and entropy counts them.

3. High-entropy macrostates have overwhelmingly more microstates, so systems drift toward them and tend to stay there.

4. This resolves Kelvin-Loschmidt: an irreversible whole from reversible parts.

// therefore, the second law is a statistical tendency, not an exceptionless law.

Statistical reversibility objection, reframed based on entropy

1. The fundamental laws of physics are time-reversal symmetric: treat past and future identically.

2. The second law of thermodynamics is not a fundamental law. It is a statistical consequence of TR laws: there are vastly more high-entropy configurations than low-entropy ones, so systems are overwhelmingly likely to be found in, and to evolve toward, high entropy.

3. Since the underlying laws are time-symmetric, this statistical reasoning applies equally toward the past and toward the future of any given moment.

// Therefore, from any moment where a system is not at maximum entropy, entropy is overwhelmingly likely to have been higher in the past as well as to be higher in the future. The entropic arrow is double-headed. This undermines our memories and consequently the truth in repeatability of scientific experiments.

Past Hypothesis Solution

1. An egg is a low-entropy system. It comes from a chicken, which is also low-entropy.

2. A chicken sustains its order by taking in low-entropy energy (food) and expelling high-entropy energy (heat). Life forms are conduits from low to high entropy.

3. The food's low entropy comes from plants, which get it from sunlight.

4. The sun is low-entropy because it formed from a diffuse gas cloud that gravity clumped together. When gravity matters, a smooth, uniform gas is a low-entropy state.

5. That gas traces back to the early universe, which was filled with a nearly uniform hot gas.

//Therefore, all the local low-entropy order we encounter (eggs, chickens, ice cubes, books, freezers, people) is traceable to the extraordinarily low-entropy state of the early universe. The arrow of time therefore originates in the initial conditions of the universe, not in the laws of physics. Why did the universe begin in such a special, highly ordered state?

Carrol's objection to Past Hypothesis Solution (1. Not truly explanatory)

1. Dynamical laws are tested at every moment. A boundary condition is imposed once, like an empirical fact.

2. 'The early universe had low entropy' and 'it is a law that it did' don't differ substantively.

3. Penrose's Weyl curvature hypothesis builds in time asymmetry by hand and explains nothing else.

//Therefore, The Past Hypothesis is a brute fact or an ad hoc law. At best it is a placeholder for a deeper principle.

Carrol's objection to Past Hypothesis Solution (2. posits an extremely unlikely state)

1. A high-entropy disordered state is overwhelmingly more probable than an ordered start.

2. So 'everything arose by a statistical fluke' is probabilistically far more likely than a low-entropy big bang.

// Therefore, The Past Hypothesis needs an extraordinarily improbable beginning, and the puzzle just shifts to cosmology.

Carrol's alternative to Past Hypothesis Solution (Boltzmann brains)

1. In an eternal universe fluctuating around equilibrium, small fluctuations are far more common than large ones.

2. If all you need is one observer, the cheapest fluctuation is a lone brain.

3. So the overwhelming majority of observers would be Boltzmann brains, and everything else would be in equilibrium. That's not what we observe.

// Therefore, The fluctuation alternative is falsified, and global skepticism is cognitively unstable.

substantivalism vs. relationalism

Substantivalists say space can exist independently of material objects; relationists say space is only a system of relations between objects

Newton's bucket dilemma

Bucket of water spinning by a string

1. Bucket spins, water still: surface flat.

2. Friction spins the water too: surface concave.

3. Rope re-twists and slows the bucket, but the water keeps spinning: surface stays concave.

The shape of the surface of water is not in relation to the bucket spinning but to what?

Newton's reply to the bucket

The shape of the surface of the bucket is in reference to 'space itself,' absolute space: a real, unmovable entity that gives the truest reference for rest, motion, and especially acceleration, which we cannot touch or perceive but which has observable consequences. Which turned a philosophical dispute into scientifically verifiable data: absolute space has observable consequences

Mach's reply to the bucket

1. Newton jumped from the lab to empty space. In the lab, there are always walls, the Earth, and distant stars to serve as reference.

2. In a truly empty universe, spinning and not spinning would be indistinguishable (no force felt, rope slack). With a few stars, a tiny force. With the matter of our universe,

the full force.

// Therefore, the force felt in the spinning bucket comes from a collective, proportional influence to all the other matter in the universe. Only relative motion and relative acceleration matter.

Spinning globes (Maudlin) / two rocks (Greene)

In an otherwise empty world, two globes yoked by a cord (Maudlin), or two rocks tied by a rope (Greene); (Newton) if they revolve there is tension in the cord and the rate and sense of revolution could be determined, so rotation must be relative to absolute space

Newton on the two rocks/spinning globes

If they revolve about their common center, there is tension in the cord, and experiments (applying forces to faces of the globes) would reveal both the rate and the sense of revolution, even with no reference body. So rotation must be relative to absolute space.

Coordinate systems and the relationist (Maudlin): what is the freedom in embedding relations into a fictional spacetime?

Arbitrary conventions such as origin and orientation (translations, rotations, parity), not real indeterminacy

For the Minkowski case, add Lorentz transformations and time inversion. The readings do not develop coordinate systems beyond this and Huggett's discussion of reference frames.

Invariant vs. non-invariant features of space: which feature is NOT detectable in classical mechanics, and why?

Absolute (uniform) velocity: the laws hold in every frame moving at constant velocity relative to one another, so inertial effects do not reveal it

Huggett: inertial effects reveal the magnitude of acceleration but not velocity (Newton's 5th Corollary; 10 mph feels like 100 mph). Maudlin: in Minkowski spacetime the invariant interval is the relevant relation.

Reference frames (Huggett): what is an inertial frame, and how are they related?

A frame in which the first law holds; there is a whole family of inertial frames moving at constant velocity relative to one another

Objects accelerating in one inertial frame accelerate in all of them. This equivalence is at the root of the undetectability of absolute velocities. (Spacetime diagrams themselves are not in the readings; Maudlin uses world-lines and simultaneity hypersurfaces.)

Classical spacetime structures (Maudlin): what does neo-Newtonian spacetime have that distinguishes it from Newtonian absolute space?

A spatial metric only on simultaneity hypersurfaces plus an affine connection (which trajectories are straight/inertial): absolute acceleration is defined but absolute velocity does not exist

This lets the substantivalist evade the kinematic Leibniz shift. A neo-Newtonian relationist would need a collinearity predicate col(x,y,z) but lacks the resources to predict inertial effects with occupied points only.

What does absolute space imply about velocity, and what is the catch?

It gives a meaning to absolute rest and absolute velocity, but absolute velocity is undetectable; neo-Newtonian spacetime keeps absolute acceleration while dropping absolute velocity

The labels 'Aristotelian' and 'Galilean' do not appear in the readings; this card covers the related material. Huggett notes Newton's inference from absolute acceleration to absolute velocity is mistaken (deferred to his ch. 10, not provided).

Bucket and globes in spacetime (Maudlin): how does the Minkowski relationist explain a rotating bucket?

Particles near the axis trace longer spacetime 'lengths' (more proper time) than those at the edge because of time dilation, so the relations differ from a non-rotating bucket; this makes Mach's program moot for the relationist

In GTR, however, the relationist with particles fails (e.g., an unrecorded gravitational wave) and must adopt a plenist field ontology.

What are the two typical strategies in the substantivalist/relationist debate?

Absolutist: find a phenomenon unexplainable without absolute space (bucket, handedness).

Relationist: show absolute space has unacceptable consequences, e.g., undetectable differences (Leibniz shifts)

Static Leibniz shift: what is it, and how is it used against absolute space?

Move the whole material universe to another absolute place (or time, or flip orientation) with no change in relations; PSR: God had no reason to choose; PII: indiscernible universes should be one

Replies to the Leibniz static shift (Maudlin)

It can be formulated only with indexicals ('here,' 'now'), so the worlds are only qualitatively indistinguishable and the PII does not apply; the PSR worry disappears once God is removed; Huggett also finds the God-based reasoning unconvincing since Clarke says God's will suffices

Maudlin adds: the Newtonian relationist treats different embeddings as the same state, and in GTR realistic spacetimes are not homogeneous or isotropic, so the static shift cannot be formulated.

Kinematic Leibniz shift: what is it, and why is it different from the static shift?

Give the whole universe a fixed extra absolute velocity with no change in relations. Absolute velocity can be specified without reference to any object, so it is a real but empirically inaccessible property; the Principle of Identity of Indiscernibles (or the maxim against undetectable facts) has bite, while the Principle of Sufficient Reason does not (isotropy leaves zero as the only non-arbitrary velocity)

Replies: the absolutist can deny the PII, or adopt neo-Newtonian spacetime (no absolute velocity). Clarke's ship

reply to Kinetic Leibniz shift

in a sealed ship, being at rest and moving smoothly differ undetectably; Leibniz answers the Principle of Identity of Indiscernibles requires only observability in principle (a shore line), which is absent for the whole universe.

The Leibniz-Huggett dynamic shift: what is it and why is it a problem for the relationist?

Change the absolute acceleration of everything without changing relations; new inertial effects appear, so the universes are discernible and the relationist cannot explain the effects

Kant's handed objects: define congruent vs. incongruent counterparts, with examples.

Congruent: the mirror image can be moved onto the original by rigid motion (letter E, sphere, cube).

Incongruent: it cannot (letter F, left and right hands). Handedness (chirality) is tied to incongruence; effs are leffs or reffs

Key fact: mirror images have identical intrinsic distance relations, so internal distances cannot distinguish them.

The qualification on incongruence (Huggett): why isn't F's mirror image incongruent in every space?

Leaving the plane (folding through a third dimension) or traveling around a nonorientable space like the Möbius loop makes it fit, so congruence depends on the global shape of space; the fix is a local definition (nearby figures not superposable by rigid motions in a sufficiently close region)

By analogy, in three dimensions a left glove could return fitting a right hand after a trip around a nonorientable space, or via motion through four dimensions.

Kant's glove argument

1. Left and right hands are incongruent, hence really different;

2. Identical internal spatial relations; the difference is external-spatial

3. The relationist can appeal only to distance relations between objects

// Therefore, relationism cannot account for it, while absolute space can.

Huggett compares it to the bucket: Newton infers absolute space abductively, while Kant's 'geometer's proof' claims a stronger conclusion, that relationism is impossible.

Replies to Kant's glove, Huggett: what are the relationist's fitting reply and the objection to it?

Handedness is relational, since a leff fits into sentences of Latin letters and a left hand fits left gloves.

Objection: reflect everything and the reff fits the mirrored sentence, so the 'fitting' theory says handedness does not change; and a lone hand has nothing to fit with. Huggett is at a loss about whether the whole universe's mirror image reverses handedness

Huggett adds the argument may backfire like the static shift: the difference between a lone left-handed and lone right-handed universe is unobservable, so the PII would make them identical.

Verdict: Kant has not shown a phenomenon the relationist cannot account for.

What is the Law of Light, and why does it matter? (Maudlin)

1. Every ray of light in a vacuum has the same speed, c, in all inertial frames.

2. The speed is independent of the source's motion (Sam's and Sue's light pulses run "neck-and-neck," unlike their bullets).

3. It is also independent of the frame in which it is measured.

// Therefore, the speed of light is an invariant quantity. The theory's fundamental feature is what it makes absolute, not what it makes relative. Light's path must also be fixed by structure in spacetime itself, not by a material medium.

Light at the same speed in all inertial frames, some immediate consequences

light travels at same speed in all inertial frames

need to give up Galilean view

Lorentz transformations

Why give up the Galilean view? (Maudlin)

1. For the Law of Light to hold, Δx/Δt must stay unchanged for any pair of events on a light ray.

2. Under a Galilean transformation, Δx changes while Δt stays invariant.

3. Under a Galilean transformation, no velocity is invariant.

// Therefore, the Galilean transformation can't represent the transition between inertial frames, so the spacetime structure implicit in it must be rejected.

Where do the Lorentz transformations come from?

1. At low speeds Newtonian physics is nearly right, so the new transformation must reduce to the Galilean one. It therefore has the form x′ = γ(x − vt).

2. By symmetry, x = γ(x′ + vt′).

3. A light ray satisfies x = ct and x′ = ct′ in both frames. Combining these gives γ = 1/√(1 − v²/c²).

4. A similar argument gives t′ = γ(t − vx/c²).

// Therefore, the Lorentz transformation is t′ = γ(t − vx/c²), x′ = γ(x − vt), y′ = y, z′ = z. Elapsed time between events is not invariant. (D adds that γ ≈ 1 at everyday speeds, which is why it took so long to discover these effects.)

further consequences, non-invariant features: simultaneity

1. time dilation: t′ = γ(t − vx/c²), so events with t′ = 0 are those with t = vx/c².

2. The set of events simultaneous with P in the primed frame is therefore a tilted line, not the horizontal t = 0 line (M, Figs. 2.12-2.14).

// Therefore, simultaneity is relative to a reference frame. P and Q can be simultaneous in one frame while P and R are simultaneous in the other. (simultaneity requires dividing spacetime into spaces at times, and there are many ways to do so.)

further consequences, non-invariant features: Temporal separation between events

1. Light travels at c in every frame (D).

2. In a frame where the light clock is moving, the light travels a longer diagonal path.

3. Because the speed is the same, the trip takes longer.

// Therefore, duration is frame-dependent: Δt = γΔt′. At 0.99c, one second in the rest frame is just over seven seconds in the other. (M reaches the same result from the Lorentz transformation, which does not preserve Δt.)

Further consequences, non-invariant features: Time order of events outside the light cone

1. Events inside P's light cone (absolute future or past) are in the same time order in all Lorentz frames.

2. For Q outside P's light cone, one frame has P before Q, another has P after Q, and another has them simultaneous (M, Fig. 2.16).

3. (D) Such events are spacelike separated: too far apart in space and too close in time for a signal to connect them. In the satellite/spaceship case, the tail receives the flash first in one frame and the nose first in the other.

// Therefore, There is no objective fact about which of two spacelike separated events came first. Timelike and lightlike related events cannot be reordered.

Further consequences, non-invariant features: Spatial separation between events

1. x′ = γ(x − vt), and γ grows without bound as v approaches c.

2. Under rotation of Euclidean axes, Δx and Δy change but the distance doesn't. Frames likewise split spacetime separations into spatial and temporal parts differently.

// Therefore, The spatial distance between two events is not invariant, and it becomes larger under the Lorentz transformation than under the Galilean one.

Further consequences, non-invariant features: Length contraction

1. An object's length in a frame is the spatial separation between simultaneous events at its two ends in that frame.

2.: Frames disagree about simultaneity, so they pick out different pairs of end-events (A and B versus A and C in M's Fig. 2.17).

3. Solving for the unprimed coordinates of C and transforming gives a length of d/γ in the frame where the bar moves.

// Therefore, a moving object is shorter than at rest, and each frame judges the other's objects shorter. This is not a dynamical squashing, because speed has no absolute status. (D: l = l′/γ, so a 50 ft train at 0.9c is about 22 ft in the tunnel frame.)

Further consequences, Length contraction example: Car/train and tunnel

1. In the tunnel's frame, the car/train is contracted, so there is an instant when both its ends are inside.

2. In the car's frame, the tunnel is contracted, so the front has emerged before the rear has entered.

3. "The car fits" means front and back are inside at the same instant, and frames disagree on which events are simultaneous.

// Therefore, there is no contradiction. Whether the doors close simultaneously is frame-dependent, and the doors don't shatter in either frame. Maudlin's moral: seemingly non-temporal notions can be "infected by hidden temporal aspects."

Further consequences, non-invariant features: Speed of anything other than light

1. Under the Lorentz transformation, no velocity other than light's is invariant.

2. (Demarest) There is no such thing as an absolute velocity, only relative velocities.

3. (Maudlin) Sublight velocities are not real properties of objects.

// Therefore, there is no objective fact about how fast anything is going, and no dynamical effect of "traveling near the speed of light." You are right now traveling at 99.99% of light speed in some legitimate inertial frame. Things don't shrink or slow down. There are only infinitely many equally valid ways of dividing spacetime intervals into space and time.

Invariant quantities: The speed of light, c

The Law of Light: Δx/Δt = c for every light ray in every inertial frame.

Almost every other quantity (distance, duration, speed, and time order for spacelike pairs) varies between frames.

Since frame-dependent quantities aren't objective facts, the invariants are where to look for objective facts about the world.

Invariant quantities: The spacetime interval

1. c²Δt² − Δx² − Δy² − Δz² has the same value after a Lorentz transformation (proven algebraically).

2. This is analogous to Euclidean distance √(Δx² + Δy²), which is preserved under rotation even as its components change.

//Therefore, the interval is the "true spatio-temporal structure," only arbitrarily divided into space and time by coordinate frames (Minkowski spacetime).

Invariant quantities: Light cones and lightlike/timelike/spacelike separation

1. The interval between an event and any event on its light cone is zero, and all frames agree on this.

2. A zero interval means lightlike separation, a positive interval means timelike, and a negative interval means spacelike.

3. (D) Equivalently, if a signal from A can reach B, they are timelike or lightlike related.

// Therefore, the light cone structure is frame-independent. Timelike or lightlike events have an objective order, while spacelike ones don't. For spacelike pairs some frame makes them simultaneous, and for timelike pairs some frame makes them co-located (M).

Invariant quantites: Inertial vs. non-inertial trajectories

1. (Demarest) An inertial path is free from external forces and is a straight line in spacetime.

2. Spatiotemporal length is given by the interval, so elapsed time along a timelike path is its "length."

3. (Demarest) Because the interval is the square root of a difference (Δt² − Δr²), straight paths have the longest length between two points.

// Therefore, an inertial path accumulates the most elapsed time between two events. A bent or non-inertial path accumulates less. (Demarest stresses that it's the path's shape, not acceleration as such. Maudlin's excerpt stays mostly within inertial frames.)

The twin "paradox" and consequences

1. At any point during the separation, each twin correctly judges the other to age more slowly.

2. The twins' situations are not symmetrical. Maggie's path is inertial (straight), while Peggy travels two different inertial paths (out and back).

3. A straight path through special relativistic spacetime has the most elapsed time.

4. In Peggy's simultaneity slices, Maggie ages slowly but discontinuously: just after the turnaround she judges Maggie to be suddenly much older (Fig. 30.5).

// Therefore, Peggy returns younger. There is no contradiction, since the symmetry assumption is false. It is her path, not her acceleration, that makes her age less. Both twins agree on this once light signals carry the information. (M's excerpt does not discuss the twins, but his clock postulate supplies the theory behind the argument.)