Great question! Let’s break this down with intuitive reasoning and a touch of topology—no overly fancy math, just common sense about how temperature and pressure behave.

First, let’s clarify the problem to make sure we’re aligned:

• At the start (t=0), we have an antipodal pair of points (let’s call them Point A and its opposite Point A’) where both temperature and pressure match our starting point’s values (let’s fix these as T₀ and P₀).
• Digging through the Earth takes time, so as we work, temperature and pressure across the entire surface change continuously—weather doesn’t jump from 20°C to 100°C without passing through every temp in between, same for pressure.
• The core question: at every moment during the dig, does there exist some pair of points (likely antipodal, echoing the initial setup) where at least one point has exactly T₀ and P₀?

Short Answer: Yes, it’s guaranteed. #

Here’s why, broken into two easy-to-follow parts:

1. Continuous changes can’t erase the initial values entirely

Earth’s surface is a closed, finite sphere—no edges, no gaps. Temperature and pressure are continuous functions over this sphere, meaning they can’t suddenly skip values.

At t=0, we know at least two points (A and A’) have T₀ and P₀. Suppose for a second that at some later time t₁, no point on the entire planet had T₀ and P₀. That would mean:

• Either every point’s temperature is strictly above or strictly below T₀—but that’s impossible, since we started with points at T₀ and temps change continuously. You can’t go from having T₀ to never having it again without a jump, which doesn’t happen in nature.
• Or, temps hit T₀ somewhere, but pressure never matches P₀ at those same spots. But the set of points with T(t,x)=T₀ is a closed curve (or set of curves) on the sphere, and pressure varies smoothly along that curve. Since we started with points on that curve where pressure was P₀, continuity ensures the pressure along that curve will still cross P₀ somewhere at every moment.

2. Even stronger: There’s always an antipodal pair with matching temp/pressure (and one might be our T₀/P₀)

A classic topology result called the Borsuk-Ulam Theorem tells us that for any continuous function mapping points on a sphere to pairs of numbers (like temp and pressure), there’s always at least one antipodal pair where both points have identical temp and pressure—just like our initial A and A’.

While this pair might not always include a point with T₀/P₀, the earlier reasoning already guarantees there’s some point on the sphere with T₀/P₀ at every moment. That point is part of an antipodal pair (every point has an opposite), so we can always say there exists a pair of points where one of them matches our starting temp and pressure.

Quick Recap #

You can’t get rid of the exact temperature and pressure you started with on Earth’s surface—continuous, smooth changes over a closed sphere ensure those values will persist somewhere at every moment. Whether it’s part of an antipodal pair or a random spot, it will always exist.

内容的提问来源于stack exchange，提问作者William Grannis