I was going to post the following as Deep Thoughts XLIII, in a series of posts meant to be largely tautologous or at least trivial statements:
- Everyone older than you was once your age.
And then I realized that this is not actually a tautology. It might not even be true.
Suppose time is discrete in an Aristotelian way, so that the intervals between successive times are not always the same. Basically, the idea is that times are aligned with the endpoints of change, and these can happen at all sorts of seemingly random times, rather than at multiples of some interval. But in that case, (1) is likely false. For it is unlikely that the random-length intervals of time in someone else’s life are so coordinated with yours that the exact length of time that you have lived equals the sum of the lengths of intervals from the beginning to some point in the life of a specific other person.
Of course, on any version of the Aristotelian theory that fits with our observations, the intervals between times are very short, and so everyone older than you was once approximately your age.
One might try to replace (1) by:
- Everyone older than you was once younger than you are now.
But while (2) is nearly certainly true, it is still not a tautology. For if Alice has lived forever, then she’s older than you, but she was never younger than you are now! And while there probably are no individuals who are infinitely old (God is timelessly eternal), this fact is far from trivial.
2 comments:
Assignment for ChatGPT:
"Prove or disprove the statement "Everyone older than you was once your age." based opun the four dimensional spacetime with the minkowski metric.
If you can't prove or rather are disproving that statement, then would you be so kind as to find a sufficiently close enough and appoximate covariant statement, which is true and provable in the four dimensional spacetime with the minkowski metric?
Thank you."
ChatGPT's answer:
"This is a very interesting and subtle question, blending everyday intuitions about age with the physics of four-dimensional spacetime under the Minkowski metric (special relativity).
---
## Statement:
> “Everyone older than you was once your age.”
Let’s unpack and analyze this under special relativity (flat spacetime, Minkowski metric).
---
## Informal Interpretation:
In Newtonian/classical mechanics, the statement is trivially true:
- Time is absolute.
- If Alice is older than Bob, then at some moment in the past, Alice had the same age Bob has now.
But Minkowski spacetime is not Newtonian:
- Time is relative to observers (due to time dilation).
- Simultaneity is not absolute.
- “Age” = proper time along a worldline — not coordinate time.
---
## Reformulating the Statement Relativistically:
We must define:
- Your age = proper time τ_B along your worldline.
- “Someone older than you” = observer A whose proper time τ_A is greater than τ_B as measured now (from your frame).
- “Was once your age” = There exists a point on A’s worldline where her proper time equals τ_B.
The statement now becomes:
> For every timelike-separated worldline A with proper time τ_A > τ_B, there exists a proper time τ'_A < τ_A such that τ'_A = τ_B.
### But this assumes too much.
Because of relativity of simultaneity, and the lack of absolute global time, it's not generally meaningful to compare ages across spacetime in this way without specifying a frame.
Even more critically:
### ❌ Counterexample (Disproof):
Let’s consider two observers:
- Bob stays at rest on Earth.
- Alice travels at relativistic speed, turns around, and comes back (classic “twin paradox”).
At reunion:
- Alice is younger than Bob.
So Bob (the stay-at-home twin) is older than Alice at reunion.
But Alice never was as old as Bob is now — and never will be.
Hence, even though Bob is older than Alice, there was never a point on Alice’s worldline where her age equaled Bob’s current age.
> So: The statement "Everyone older than you was once your age" is false in Minkowski spacetime."
... ## ✅ A Covariant, Approximately True Statement
We now seek a covariant (coordinate-independent) statement, as close as possible in meaning, that is true in Minkowski spacetime.
Try this:
### ✅ Covariant Alternative:
> “If two observers' worldlines intersect and one has experienced more proper time between a shared starting event and the intersection, then that observer is older at the event of intersection.”
### Why this works:
- Proper time between two events on a timelike worldline is invariant.
- If two observers share a common event (say, birth or synchronization), and later meet again, their aging can be objectively compared.
- This avoids simultaneity issues.
Or slightly rephrased:
> "Among all timelike curves connecting two fixed events, the one that is an inertial (straight) worldline experiences the maximum proper time."
This is the essence of the twin paradox, and is provable from the Minkowski metric.
---
## ✅ Another Approximate Version:
For observers who are approximately at rest with respect to each other (small relative velocities), we can say:
> "To a good approximation, in an inertial frame where simultaneity is defined conventionally, an older observer had the same age as a younger observer at some earlier event in the past."
But this is frame-dependent and only approximately true.
---
## Conclusion
- ❌ The original statement is false in Minkowski spacetime.
- ✅ A covariant, true version would require:
-Shared events.
- Comparison of proper times along timelike worldlines.
- ✅ Approximate versions may hold in low-speed, non-accelerated contexts.
Would you like a diagram illustrating this in spacetime?"
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