Notes on Observational Indistinguishability

Bryan W. Roberts

University of Southern California

ARLT-100: Einstein's Spacetime Revolution

Background Reading: Norton on Observationally Indistinguishable Spacetimes

**1.1. Deductive inferences** in a scientific theory are inferences to the logical consequences of a theory. They are typically mathematical results that are *guaranteed* to be true whenever the assumptions of the theory are true.

**Examples of Deductive Inferences:**

- Tommy is from Troy. (How dow we know this? We assume that Tommy is a Trojan and that Trojans are from Troy -- the conclusion follows as a matter of simple logic.)
- In special relativity, the rate of clock in inertial motion with respect to some fixed frame of reference will be slower than the clocks in the fixed frame of reference. (Question: How do we know this?)
- In special relativity, whether two spacelike-separated events are simultaneous depends on which inertial observer we ask. (Question: How do we know this?)

**1.2. Inductive inferences** in a scientific theory are that are not logical consequences of the theory, but which we nevertheless are warranted to infer.

**Examples of Inductive Inferences:**

- The sun will rise tomorrow. (How do we know this? We know the sun rose yesterday, and the day before, and the day before. This makes it extremely likely -- though not a logical certainly -- that it will rise again tomorrow. In this sense, we are ``warranted'' in inferring the conclusion.)
- There are no time travelers in our near future. (How might we argue this?)
- There is 0 summed curvature above the surface of the earth.

**2.1. What can you know** about the spacetime you're in? *Answer:* You can know at most everything in your past light cone.

**More optimistically:** You can aggregate information from a number of observers, including yourself, over the course of your lifetimes. Each ``observation'' will be a description of an observer's past light cone at a moment. Like this:

**2.2. The problem** is that no matter how many such observations you have, a recent discovery due to John Manchak shows that there are always at least *two different* spacetimes that are compatible with those observations. If a spacetime is such that any collection of its past light cones can be mapped onto a distinct spacetime in a way that preserves the distance relations on the light cones, then the two spacetimes are said to be *Observationally indistinguishable*.

The most important result in this regard is *Manchak's theorem:* In spacetimes with no closed timelike curves, no number of observations is enough to know which spacetime we're in -- even assuming Einstein's equation and other local assumptions within general relativity!

Because of the way this theorem works, it is not only the case that we cannot *deductively* determine which spacetime we're inhabit within the context of general relativity. But as Norton argues, our best attempts to *inductively* determine which spacetime we're in also seem to fail as a consequence of Manchak's theorem. Can you see why?

- The difference between inductive and deductive inference and how to tell them apart.
- What observationally indistinguishable spacetimes are.
- What Manchak's theorem says.
- How it is that Manchak's theorem undermines any attempt to deductively infer which spacetime we're in.
- How it is that Manchak's theorem also seems to undermine attempts to inductively infer which spacetime we're in.