I’m sure anyone who has hung out long enough here on Physics Forums has encountered threads that go something like this (I’ll use an example based on threads I’ve seen and participated in in the relativity forum, but I’m sure similar things occur in other forums as well):
Original Poster: I don’t understand how black holes can actually exist. Doesn’t it take an infinite time for anything to fall in?
SA/Mentor: The “infinite time” is just coordinate time; if you calculate the proper time experienced by the infalling object when it reaches the horizon, it comes out finite.
[Exchange follows in which the actual math may even be shown or linked to.]
Original Poster: Sorry, I don’t understand all that math. Can’t you explain it in plain language? If you can’t put in in terms that make sense to me, I don’t believe it, no matter what your math says.
(Please note, the above are not direct quotes, and I am not going to name any names because I have no desire to single anybody out. I am simply trying to distill a common argument down to what seems to me to be its bare essentials.)
What I’m about to say is going to sound harsh, and in a way, it is harsh, which is why I’m saying it here instead of in any of the numerous forum threads in which I’ve been tempted to. Here it’s not directed at anybody; I’m just stating something that I think is true. Here it is:
Table of Contents
If you don’t understand the math, you’re not entitled to an opinion about the theory.
Richard Feynman once said, “If you want to understand Nature, you must learn the language She speaks in.” It’s all very well to try to get a start by reading descriptions in English, or whatever your language of choice is, of what a scientific theory says. But those descriptions are not the theory. You can’t form an opinion about the theory from them. You have to understand the actual theory, i.e., the math.
“But scientific theories aren’t just math.”
Yes, I know that. Obviously, the math is no good unless you have some way of linking the math up with experience. That’s also part of the theory, yes. But that doesn’t mean you can get away with not understanding the math, because (1) the math is what makes the predictions, and the predictions are numbers anyway, and (2) the data you’re going to compare the predictions with are numbers too, often numbers which require sophisticated interpretation before you can compare them with the predictions. And how do you do such sophisticated interpretation? With math.
I want to make it clear what I am not saying. I am not saying that scientists, and people like me who are not practicing scientists but who are knowledgeable about at least some areas of science, shouldn’t try to give clear descriptions in the plain natural language of what a theory says. They should. I try to do that here on PF. But these are descriptions of what the theory says, as best it can be translated from math into natural language. The OP in my example above, and many others like him, want to demand proofs in natural language that the theory is correct, and that is just not going to happen.
“No, you’re wrong. I don’t insist on a proof. But I do insist on some explanation of what’s going on that makes sense.”
Same answer: there may not be any such explanation in a natural language that “makes sense” according to your criterion. (Very often the person making this demand doesn’t realize that what “makes sense” to them already implicitly makes a lot of assumptions that are simply not true in general, however good they may be as approximations in everyday life.) Or there may be such an explanation, but nobody has thought of it yet. It can take decades even for experts in the field to understand some aspect of a theory, and they know the math. (The history of theorists’ understanding of Schwarzschild spacetime is a good example of this, one which has given rise to a fair proportion of the threads that first gave me the idea of writing this post.)
“Yes, I know your math says X. But this other math says Y, which is inconsistent with X. And Y seems much more intuitively sound to me. So I believe Y.”
(For example, X = the proper time to the horizon for an infalling observer is finite; Y = the coordinate time is infinite.) No, Y is not inconsistent with X. To someone who understands the math, this is obvious; but if you don’t understand the math and are relying on natural language descriptions, yes, they certainly can sound inconsistent, particularly since many authors are sloppy in their terminology because they are more concerned with getting across some pictures of what they’re talking about than with strict accuracy and consistency. They don’t expect what they write to be taken as a proof of the theory, or a completely consistent explanation of it, just as an attempt to describe some aspect of it in a way that is not going to be judged based on apparent consistency with other aspects.
“So you’re saying I can’t trust these authors to tell me what’s really true?”
If by that you mean “tell a completely consistent story that encompasses all aspects of the theory”, then no, you can’t. There is just no way to tell that story without the math. Here’s why: a theory is not just a description of what happens. It’s a way of generating predictions about what will happen in scenarios you haven’t looked at yet. For practicing scientists, of course, “scenarios you haven’t looked at yet” means “scenarios that haven’t yet been tested in experiments by anyone”, so a practicing physicist in General Relativity doesn’t have to spend a lot of time verifying that GR gives the correct prediction for, say, the precession of Mercury’s perihelion; he’s already been there and done that. But if you’re posting here on PF asking questions about GR, you probably haven’t been there and done that; so for you, it’s perfectly legitimate to ask how GR comes up with the correct prediction for the precession of Mercury’s perihelion (or anything else it predicts). But you can’t do that just from natural language descriptions of GR, because GR doesn’t use natural language descriptions to make its predictions; it uses math. So relying on natural language descriptions of GR to generate your predictions won’t work; you’ll be working with the wrong set of concepts.
Here’s a simple example (using SR rather than GR, but the point is the same): we get fairly frequent threads here on PF where someone is trying to figure out how “time dilation” works and getting obviously nonsensical answers. The thread will go on for many, many posts, with people trying to explain why the OP is getting obviously nonsensical answers and how they need to change how they are looking at the problem, but sometimes it just doesn’t get through. The reason is simple: the OP simply doesn’t get that “time dilation” in SR is not a fundamental concept; it’s not what the theory uses to actually generate predictions. All it is is a language that some physicists use to describe what happens, *after* they’ve already made a prediction using the actual theory (the math) and verified that it’s correct.
“But then why do all those pop-science books and TV shows give all those colorful natural language descriptions? Aren’t they trying to explain the theory to us?”
Yes and no. They’re trying to “explain” the theory, for some value of “explain”, yes; but if you’re asking questions here on PF, you’re probably not their intended audience. If you’re asking questions about a scientific theory here on PF, you’re already different from most people who read popular books or articles about science. Most people who read natural language descriptions of a theory don’t want a completely consistent all-encompassing story that they can use to generate predictions; they just want a quick “sound bite” that gives some flavor of what the theory says. Those are the people most of these popular authors are writing for. (There are exceptions, and I and others here on PF try to point to them where we can. Kip Thorne’s Black Holes and Time Warps is one example, a popular book that, while it can’t tell the complete story since it doesn’t include the math, manages to tell quite a lot of it without too much distortion in translation. But even that isn’t enough to “prove” that GR is “correct” or to use it to make correct predictions, if that’s what you’re looking for.)
If you’re posting here on PF, you are hopefully looking for more than just a quick sound bite or a nice-sounding description that may or may not match the actual theory. That’s great! Please post and ask questions. But don’t be fooled into thinking that we can paint a comprehensive, self-consistent picture of any scientific theory, that generates correct predictions and shows you how it’s done, without using the math. It can’t be done. So if that’s what you’re looking for, you’ll have no alternative but to learn math. If you aren’t willing to do that, then you aren’t entitled to an opinion about the theory. You may not like it, but that’s the way it is.
- Completed Educational Background: MIT Master’s
- Favorite Area of Science: Relativity