The Andromeda Paradox: How Walking Changes the Future

Two strangers pass each other on a sidewalk. One heading north, one heading south, at ordinary walking speed. Nothing in the scene looks unusual. But something quiet is happening, two and a half million light-years away, in a galaxy neither of them will ever visit.

According to the mathematics that guides GPS satellites and predicts planetary orbits, these two walkers do not agree about what is happening in the Andromeda Galaxy right now. Not by seconds. By roughly six days. In one walker’s mathematical present, an event over there has already occurred. In the other’s, that same event is still six days in the future.

This is the Andromeda Paradox, the vivid thought experiment Roger Penrose used in The Emperor’s New Mind to dramatize an older, cleaner argument in the philosophy of physics. It is where the block universe interpretation of relativity gets its sharpest edge, and where its deepest assumption — the one the physics does not quite force on you — is easiest to see.

This piece walks through where the six-day number comes from, why some careful physicists say it means the future already exists, and why other careful physicists, using the same mathematics, reach a very different conclusion.

What the Andromeda Paradox is

Andromeda, catalogued as Messier 31 or M31, is the nearest large spiral galaxy to our own. It is a wheel of stars roughly 152,000 light-years across, containing about a trillion suns. Measured from Earth, it sits about 2.5 million light-years away — a distance confirmed directly in 2005 by eclipsing-binary measurements at 772 kiloparsecs, plus or minus 44. Converted to metric, that is roughly 2.365 × 10²² metres, or about 14.5 quintillion miles.

Light from Andromeda takes 2.5 million years to reach us. Nobody sees Andromeda in real time. Every photon striking a telescope tonight left when there were no cities on Earth, no pyramids, no writing, no wheel.

The paradox is not about what any observer can see. It is about what each observer’s mathematical frame assigns to the label “present.”

Special relativity, formulated by Albert Einstein in 1905 and given its geometric shape by Hermann Minkowski in 1908, removes the Newtonian idea of a universal clock. Time is not a stage on which events unfold. It is a coordinate — one of four — that describes each observer’s slice through a four-dimensional structure called Minkowski spacetime. Two observers in relative motion do not carve that structure the same way. Their planes of simultaneity — the three-dimensional slices they each call “everything happening now” — are tilted relative to each other.

At everyday distances, the tilt is invisible. At the scale of a galaxy, the tilt turns into days.

The math: where the six days come from

The rule governing the tilt is the Lorentz transformation. For motion slow compared with the speed of light, the offset between two observers’ presents scales linearly with the distance to the far event and with their relative velocity:

$$\Delta t \approx \frac{v \cdot x}{c^2}$$

Plug in walking speed, one metre per second. Plug in the distance to Andromeda, 2.365 × 10²² metres. Divide by c² — 8.988 × 10¹⁶ metres-squared per second-squared. The number that falls out is 263,158 seconds. Divide by the 86,400 seconds in a day, and you get 3.05 days per observer.

Two walkers moving in opposite directions therefore disagree about Andromeda’s present by about six days. A University of Texas at Austin educational note derives the same figure. Penrose himself, in the passage that gave the paradox its name, describes the effect as amounting to “several days.”

Scale it up and the picture stays linear. A highway car at 30 metres per second — around 67 miles per hour — pushes the Andromeda offset to about 91 days per observer. A jet airliner at 250 metres per second reaches roughly 2.1 years per observer. Nothing in this calculation exceeds the speed of light. Nothing is sent, transferred, or displaced. It is a difference in how two observers foliate the same block of spacetime — how they cut the same loaf of bread at different angles.

Penrose’s dramatization

Here is the picture Penrose drew, almost verbatim from The Emperor’s New Mind, pages 303–304. Two people pass each other on the street. According to one of them, an Andromedan space fleet has already set off on its journey. According to the other, the decision about whether the fleet will launch has not yet been made.

Was there any uncertainty about that future? Or was the future of both people already fixed?

The point is not to insult the reader with a science-fiction detour. It is to show that once the tilt of a simultaneity plane is taken seriously as a real slice of what exists, the far end of that slice — which is spacelike-separated from both walkers — has to be treated as already determinate. Different walkers, different determinate futures. Both real. That is the awkward output.

The Rietdijk-Putnam argument

The formal move behind the paradox predates Penrose by more than twenty years. The Dutch physicist C. W. Rietdijk published it in 1966 in Philosophy of Science, under the confident title A Rigorous Proof of Determinism Derived from the Special Theory of Relativity. A year later, working independently, the American philosopher Hilary Putnam published Time and Physical Geometry in The Journal of Philosophy and reached almost exactly the same conclusion. Nicholas Maxwell revived the argument in 1985.

Cleaned up, the argument runs like this. Consider two observers, A and B, at the same spatial point at some moment. In A’s frame, a distant event E is happening now, so E must be as real as A. Now consider observer C, moving relative to A but passing through the same point at the same moment. In C’s frame, a different distant event F is happening now, so F must be as real as C. Since A and C are at the same point in spacetime, one cannot be real while the other is not. Therefore both E and F are real. Iterate this reasoning across enough frames and enough distances, and the conclusion drops out: every event in spacetime is as real as every other. The past, the present, and the future are all equally there.

This position has a name. Eternalism, or the block universe, or four-dimensionalism, or B-theory of time. Presentism — the intuitive view that only the present moment exists — is, on this argument, false.

Where the argument slips

The physics up to this point is uncontroversial. The Lorentz transformation is one of the most heavily tested pieces of mathematics in science. The six-day figure is a straight arithmetic output. What has not been paid for is the leap from “the labels disagree” to “each label picks out a slice of what is real.”

That leap depends on three claims, and only the first is physics:

1. For any observer at any event, there is a well-defined plane of simultaneity — a global slice through spacetime containing every event that observer calls “now.”
2. Every event on that slice is as real as the observer at the origin.
3. The reality of an event does not depend on which frame is used to label it.

Claim one is a coordinate statement. Special relativity gives it to you. Claims two and three are metaphysics. Special relativity is compatible with them. It does not require them.

The American philosopher of physics Howard Stein attacked exactly this point in two papers, in 1968 and 1991. His argument is technical but the shape is simple. “Simultaneous in some frame” is not a transitive relation. Event E can be simultaneous with F in one frame, and F with G in another frame, without E being simultaneous with G in any frame. If you use that relation as a proxy for “is real relative to,” you get a definition that lets everything be real relative to everything else — which is either a proof of eternalism or a proof that the relation is the wrong one to use.

Stein’s answer is that the right notion of “definite as of” in relativity is not the global simultaneity plane at all. It is the past light cone of a point. Everything that could have causally influenced the observer at that point is definite as of the observer’s here-now. Nothing outside is. On this reading, Andromedan events are not “in” any walker’s present. They are outside the past light cone of every walker on the sidewalk. The disagreement is a coordinate labelling, not a disagreement about what exists.

N. David Mermin, one of the most careful physicists to write on the foundations of relativity, put the point plainly in It’s About Time (2021): the single most important lesson to be learned from relativity is that no inherent meaning can be assigned to the simultaneity of distant events. Reichenbach and Bernard Weingard had already stressed that distant simultaneity in relativity is conventional — parameterized by a synchronization choice — long before the argument reached its modern form.

The Italian theoretical physicist Carlo Rovelli has been especially direct about this. In a paper titled Neither Presentism nor Eternalism, Rovelli argues that Putnam misinterprets Einstein’s simultaneity and mixes relativistic and non-relativistic concepts. In Rovelli’s reading, Einstein did not discover that there are many equally real simultaneity surfaces. He discovered that simultaneity itself has no ontological meaning beyond convention. That result destroys an absolute Newtonian present. It does not promote all events to equal reality.

Does the block need no time to pass?

Even granting a block, does it follow that the felt flow of time is an illusion? The American philosopher of physics Tim Maudlin argues no. In The Metaphysics Within Physics (2007) and his subsequent Remarks on the Passing of Time, Maudlin defends what he calls the reality of temporal passage. On his view, the passage of time is a fundamental, irreducible fact about the structure of spacetime, an intrinsic asymmetry with no spatial counterpart and metaphysically independent of the entropy gradient of the universe. Maudlin holds both eternalism and the reality of passage. The past is real, the future is real, and time genuinely passes.

Not every philosopher agrees with Maudlin. But his argument shows something important. The claim that the block universe implies no passage of time is not a straightforward consequence of the physics. It is a specific metaphysical interpretation, and one that the leading defender of passage rejects. Asked whether relativity really says the distinction between past, present, and future is an illusion, Maudlin’s answer, in an interview for IAI TV, is flat: nothing in Einstein’s theory of relativity suggests that.

The six days at Andromeda are real as mathematics. They are not automatically real as ontology. The identification of a coordinate slice with a slice of what exists is an addition to the physics, not a consequence of it.

Einstein’s two admissions

Popular treatments lean heavily on one sentence Einstein wrote near the end of his life. In March 1955, days after Michele Besso had died, Einstein wrote to Besso’s family. He told them that for people who believe in physics, the distinction between past, present, and future has only the meaning of a stubbornly persistent illusion. The line is verified in the Einstein Archives, in Freeman Dyson’s Disturbing the Universe, and in Alice Calaprice’s Ultimate Quotable Einstein. It reads, on the surface, as an endorsement of eternalism from the founder of relativity.

Read alongside another remark from the same man, the picture is less clean. In Rudolf Carnap’s Intellectual Autobiography, Einstein confided that the problem of the “Now” worried him seriously. The experience of the Now, he said, means something special for a person, essentially different from the past and the future, but that difference does not and cannot occur within physics. That the experience of the Now could not be grasped by science was, in Einstein’s own words, a matter of painful but inevitable resignation.

Two statements from the same physicist. One sounds like a manifesto. The other sounds like an admission that the framework is missing something. Both are on the record.

Quantum mechanics complicates the block

The Rietdijk-Putnam picture is a purely classical block. Every event is a fixed point in a four-dimensional manifold. But the actual matter of the universe is described by quantum mechanics, and quantum mechanics does not fit neatly into a fully-determined block.

The Schrödinger equation is deterministic. Given a wavefunction at one time, it fixes the wavefunction at every future time. In that sense the quantum world is as determined as the classical block. But the Born rule — the rule that assigns probabilities to individual measurement outcomes — introduces something the Schrödinger equation does not: probabilistic determinacy of specific results. The reconciliation of the two is called the measurement problem, and it is not solved.

Different interpretations handle the tension differently. Many-Worlds, first proposed by Hugh Everett III in 1957, keeps unitary determinism by branching the universal wavefunction at every measurement — the block becomes a bushy, ever-branching structure containing every possible outcome. Relational Quantum Mechanics, developed by Rovelli in 1996, treats quantum states as relations between systems and denies that there is a view from nowhere. The Transactional Interpretation, developed by John Cramer in 1986 and extended by Ruth Kastner, treats quantum events as atemporal handshakes between offer and confirmation waves spanning spacetime.

For our purposes the point is this. The block universe as popularly presented is a classical object. Making it consistent with quantum mechanics requires importing metaphysical commitments — extra worlds, or atemporal transactions, or relational-only realities — that go beyond the plain classical picture. The block is not a neutral consequence of physics. It is one metaphysical layer among several.

Curved spacetime, closed timelike curves, and frame-dragging

Once general relativity is added to the story, some of the strangest known solutions of Einstein’s equations become available. In 1949, Kurt Gödel presented an exact rotating-universe solution as a birthday gift for Einstein’s seventieth. His universe contained closed timelike curves — worldlines that loop back on themselves — showing that general relativity, taken by itself, does not forbid time loops.

Roy Kerr’s 1963 solution describing rotating black holes carries similar oddities inside its inner horizon. Its outer region contains the ergosphere, where frame-dragging — the Lense-Thirring effect — forces every observer to co-rotate with the black hole. In 2011, Gravity Probe B, launched by NASA and Stanford in 2004, measured Earth’s frame-dragging drift at −37.2 milliarcseconds per year against a general-relativistic prediction of −39.2, and the geodetic drift at −6601.8 against a prediction of −6606.1. Space really is dragged by a rotating mass. That is not a philosophical claim.

Igor Novikov, working with Kip Thorne and colleagues, added the self-consistency principle in 1990: on a closed timelike curve, only histories that are consistent with themselves can occur. Paradox-generating trajectories are excluded by the structure of the physics.

The end of time — Barbour’s Platonia

Julian Barbour takes the eternalist impulse to its logical conclusion. In The End of Time (1999), and in decades of subsequent work, Barbour argues that time itself does not fundamentally exist. Reality is a static configuration space he calls Platonia, in which each point is a complete, instantaneous “Now.” The apparent flow of time is emergent — a feature of what he calls time capsules, records like the tracks in a cloud chamber that contain within themselves the internal evidence of an ordered past.

Barbour anchors his picture in the Wheeler-DeWitt equation, an attempt by Bryce DeWitt and John Wheeler in 1967 to combine general relativity with quantum mechanics. Its striking feature: the equation contains no time parameter. Taken literally, it describes a state, not an evolution.

Whether Platonia is right is contested. Barbour himself concedes, on his own website, that the Wheeler-DeWitt approach remains controversial. But it sits at the far end of a spectrum the Andromeda Paradox opens up. Once the felt “Now” is denied any privileged status, it is a small further step to say the flow of time is not fundamental at all.

Where the honest reading lands

The mathematics gives us a six-day offset at Andromeda’s distance for two walkers moving in opposite directions on a sidewalk. That number is real, in the sense that it falls out of one of the best-tested equations in physics. It becomes controversial only when it is used to argue that the future is already fixed.

The strong argument — Rietdijk’s — smuggles a metaphysical claim into the geometry. Howard Stein’s rebuttal is technical and careful, and many working philosophers of physics find it decisive. Reichenbach and Weingard remind us that distant simultaneity is conventional. Mermin says no inherent meaning can be assigned to it. Rovelli explicitly rejects Putnam’s eternalism while equally rejecting presentism, arguing that reality is exhausted by relations between systems. Maudlin defends the reality of temporal passage from inside the block. Even Daniel Dennett, working from a naturalist framework, has argued that determinism, if it holds, does not entail the elimination of meaningful choice.

The deeper open question — the one both sides of the debate face honestly — is what the felt “Now” actually is. Einstein admitted it worried him. The physics has no equation for it. And the honest reading of the Andromeda Paradox is that it reveals a limit in the ordinary notion of a universal present without deciding, on its own, what the correct picture of reality replaces it with.

The block universe is one interpretation among several. The Andromeda Paradox does not settle the argument. It is a number, wrapped in a thought experiment, that points to a question the mathematics can pose but cannot answer.

Which is, in the end, what the interesting parts of physics tend to do.