# Near-relativistic car speeding over a bridge

I have read this problem / exercise (I don't remember the exact numbers, so I put my own) in William Burke's "Applied Differential Geometry" and it has always puzzled me regarding the relation between inertial and gravitational mass:

A car with a rest mass of 1000kg is speeding at 0.9c over a bridge, which is able to bear loads of up to 2000kg. Is the bridge going to collapse?

Proposed solution:

According to special relativity, the inertial mass of the car with respect to the rest frame of the bridge will be 2294kg. If, according to the equivalence principle, inertial mass equals gravitational mass, shouldn't the bridge collapse?

• Everything within roughly a mile of the park is leveled, and a firestorm engulfs the surrounding city. So the bridge would certainly collapse. – rob Aug 31 '18 at 8:43
• @rob: What if there were no air resistance? – user7777777 Aug 31 '18 at 8:46
• @user7777777 The linked analysis (firestorm, death, etc.) also applies if the car touches the bridge. If the car doesn't touch the bridge, it's hard to see why it would collapse. ... If the question really is about whether the inertial mass $m$ or the total mass-energy $\gamma m$ determines the "weight" to be supported, you might imagine instead parking a 1000 kg flywheel on the bridge and then spinning it up so that its rim is moving near $c$. – rob Aug 31 '18 at 9:02
• @rob: "Gedanken"-experiments in relativity are idealized and are meant to probe the principles of the theory. Almost all of them, including Einstein's, would collapse, if we were to assume them taking please in a realistic experimental framework. However, I assume that you understand the point that Burke was trying to drive home. – Bryson of Heraclea Aug 31 '18 at 9:11
• This question really boils down to two things : (1) are gravitational and rest mass the same and (2) is relativistic mass "real". I'd direct you to this question which itself links to a few others. – StephenG Aug 31 '18 at 12:48

The question now becomes: how much will the car fall during the jump? In the free-falling frame, the car will not fall at all: it will move horizontally, as in this frame gravity is absent. So, reverting to an earth-bound frame, we can safely assert that the car will have a vertical acceleration equal to $g$ (independent of its mass and speed). Let $s$ be the bridge span: then the car will fall $h=(g s^2)/(2 v^2)$.
Just for fun, let's put some numbers. Speed $v$ is given; as to $s$, the longest suspension bridge on Earth does not reach 2000 m, and the longest one under construction barely exceeds this length. So we may take $s=2000\rm m$. We find $h=2.7\times10^{-10}\rm m$: the size of an atom. The driver would hardly care of such a drop.
Anyhow, to answer the question: if you want to cancel the drop, you will have to apply a vertical force causing an acceleration opposite to gravity. The SR formula relating force to transverse acceleration is $F=m\gamma a$, and the force the bridge must apply to the car to sustain it is $m\gamma g$, greater than the maximum load the bridge can bear.
Would it mean that the bridge would collapse? Not at all! The maximum load is a static one, whereas the stress due to the car would last $s/v=0.74\mu\rm s$. Surely the bridge would not even notice the passage of the car.