Old question.

I'm no scientist or mathmatician but couldn't you just weigh a know object?

If you know, for example, that a brick weighs 3 pounds and the next day it weighed 1.5 pounds you would know that it shrunk even though it appears to be the same size based on its realtionship with other things.

Specific gravity (density) would increase.

(by x8)

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Hi Boomerang,

Weight is something I did not consider, but I will tell you why I was thinking this way. I can foresee something shrinking in size but keeping all it's original mass, still weighing the same but becoming more dense. Or possibly weighing less, but maybe not exactly half the original amount. That is a great point though, I will have to think about it some more. Thanks!

oh and Steve, I posted to fast to see your reply as well, thanks!

Do you mean that weight would not be a good indicator like I mentioned in my above post? An object *could* shrink and become more dense/compact still weighing the same and meeting the dimension requirement? Just trying to look at this from all angels I can. Simply curious. Thanks again for all the replies!

I have to guess the 50s book is not using the speed of light, which, of course, is a constant, as an argument.

But...
If you take an object and reduce all if it's dimensions by one half, the cross-sectional area is reduced by one fourth, and the volume is reduced by one eighth. In this case mass would also be reduced by one eighth. The strength of the object is reduced by one fourth, since strength is directly proportional to cross-sectional area. So we have an object which is supporting one eighth the mass but is only reduced in strength by one quarter. Proportionally, the object is twice as strong as the one twice the volume.

Since a meter would really be 50 centimeters and a kilogram would really be 500 grams, while strenght hasn't diminished as much, we'd see a lot of world and olympic records being broken.

On an aside, if only size changes (it doesn't matter if our ruler got small too), and mass does not change, then there has to be a change in density.

Hamel

If the dimension halves, the volume decreases by a factor of (1/2)^3 i.e. 1/8.

As the mass does not change (unless you were assuming all dimensions in the broader sense, i.e mass, space-time and length all change) you have the same mass in 1 eighth the volume so the density goes up by a factor of x8.

But of course you can't measure the dimensions of the brick halving over night, because your measuring stick halves in length too.

But then to argue against myself a moment....the aparant volume would not change the number of units of length on each side of the brick for instance and would be the same as before. And the mass does not change, so the density would be measured as the same.

So this wouldnt work after all.

So er er thinks...the trick must be to measure some physical attribute that depends on a non linear relation to the distance. Eg electromagnetic or gravitational attraction. The force of gravity is proportional to the inverse square? of the distance between the objects. Therefore if the distance halves the gravitational attraction between the brick and the cg of the earth will increase. Your brick would weigh more, but its specific gravity would be measured as the same.

This is quite an interesting mind experiment. I hope someone can show its not just as simple as weighing a 3 pound brick!

I assumed Hamel was specifically refering to the dimension of length. You were werent you Hamel?

Mass and space-time remain as before correct?

Wow, Laeknir Scrat that is fantastic!

I was pretty convinced that there was a more reasonable answer to this question. You are 100% right on that the 50's book did not mention anything of the sort I did and this must have been what they were getting at. I see I need to keep studying math to really be able to spot these things well. I'm getting there slowly. Anyway thank you for the answer here as this would be much more practical than trying to measure light travel time!

Steve, yes I am coming from the view that it is the spatial dimensions it is concerned with. That the mass stayed the same - every objects density would increase. I may have misunderstood the question - but that is how it is stated in the book. Still all of these ideas are awesome, everything you guys are saying is great for brainstorming. At first I didn't think this question was all that big of a deal but as I started to think about it I got really curious as I could not see a simple answer.

Also, sorry for the slow replies. I am at work trying to get back here as much as possible. Thanks again!

Also from Laeknirs Scrat's reply look like we can figure this out whether the mass changes or not. You guys are great!

the mass would not change full stop.

the weight would increase because of the greater gravitational pull of the earth.

I checked out the copyright on the text now that im home, I was a little off, it was 1960 but no matter. The title was "Physics For Students of Science and Engineering" authored by David Halliday and Robert Resnick just in case anyone was interested.

Steve, I have a small question as I might have misunderstood your last statement. I thought that only the tidal part of the gravitational field would change if say the Earths mass was squashed into a sphere half the diameter it's former value.
My understanding is basic so forgive me if this is way off!

If the mass did not change then the total gravitational pull would be the same at any distance, but the tidal change or rate that gravity increased/decreased moving closer or further away would be much less gradual.

I say this because I have read for instance that a Blackhole has no more gravitational pull than the star that it was born from(theoretically). That even though the mass is squashed into much much less space the total gravitational pull would be the same - just the tidal effect, the rate of change in the gravitational pull over distance, would increase dramatically.

The mass of the brick or the earth would not change. The gravitational pull between earth and brick can be considered as between the two points, the centres of gravity of the two objects.

It is a product (if my memory of 30+ years agoserves correctly) of the two masses, the Universal Gravitional constant, and the inverse of the square of their distance apart.

The distance apart would appear the same because as I keep saying, a simple linear measuring device would also have shrunk in size, thus giving the same number of units of length seperating the two centres of gravity.

But in fact the objects are nearer together. And as far as I can figure out, if you halve the distance you multiply the force by 4.

The gravitational field around a black hole is so strong that immense differences in forces apply even along a body's own length. Hence matter is ripped apart by the differential gravitational tug between the point of the matter nearest the black hole and those further away.

I'm sure there are lots of ways that the halving of dimensions could be spotted, measuring a Universal constant such as the speed of light is one, but I have a sneaking wish for someone to demonstrate that we could never know!

Okay so just going back a second here.

When I first considered this question I tried to picture myself waking up in a world where all dimensions of height, length, and width are halved including my own dimensions(my body) and all tools of measure. So proportionally all would look the same. All would measure the same.
Completely ignoring the problem of volume.

A container whos length, height and width are halved could not hold 1/2 the volume of the original container only 1/8. Therefore the container would burst if sealed. I think I was making this harder than it was intended.

Picturing the problem I was thinking that because you would have no way to tell the containers dimensions were halved by looking at it, it looks proportionally the same as yesterday, and volume is based on those dimensions - measuring the length, height and width with your shrunken ruler you would just come to the same total as the day before. In reality though all containers that were full and sealed would burst open. Or on a more positive note my trucks gas tank at 1/8 yesterday is now full and I don't have to stop on my way to work!

This is why I cried "eureka" in an earlier post, just didn't have time to write it all up. I like this question as it takes the math off the page and makes you really think about what would happen.

I still think that measuring the time it takes light to travel the distances would be another good way to prove or disprove. Just not at all practical.

And I would bet there are more ways of looking at this to refute the statement. Thinking.

Steve - i'm definitely thinking about what you have written. My knowledge of the laws of gravity are pretty limited so i'm thinking some more reading is in order for myself.
I really appreciate you taking the time to explain though, what you say does make sense.

I think you have to assume that the contents of the containers would shrink as well. Therefore, the sealed containers would not burst. After all, you're body is a container...

The speed of sound is an alternative to the speed of light that would be easier to measure. It's not constant, but you sure would notice if it doubled!

True, I did consider that our bodies are containers themselves. You are right though no reason to go into it further if we wouldn't survive ha ha. So you do NEED to assume volume shrinks to fit the new containers. Okay.

Sound. That makes a lot of sense, I never even thought of it. You are right, it doesn't matter that it's not a constant. You would notice if it's speed doubled. Thank you Markr!

Also I pointed out the text as this is the only version of the question I can get my hands on. I know I have seen it before though elsewhere so im kinda guessing that they are not the ones who came up with it. That it is probably a version of the question passed down from long before.